Tamil guerrillas have seized control of much of the northern quarter of this Indian Ocean island nation, and officials here are beginning to concede that government security forces are barely able to move beyond their garrisons.

Sustaining an almost continuous hit-and-run offensive that began in November, the guerrillas have extended their attacks to the eastern provinces near the strategic port city of Trincomalee, straining Sri Lanka's 11,000-man Army almost to the breaking point.

While official pronouncements confidently forecast a wearing down of the insurgency, privately officials show signs of discouragement and frustration over the Army's failure to confine the guerrilla war to the isolated Jaffna Peninsula at the northern tip of the island.

The key to the recent successes of the longstanding insurgency, Sri Lankan officials said, is an unbroken supply line to the Northern Province from guerrilla redoubts along the Indian coastline, just 18 miles across the narrow Palk Strait. Despite the deployment of virtually all the Sri Lankan Navy's 27 patrol boats along the northern coast, the blockade has been only partially effective in inhibiting the flow of arms and men, officials said.

Although Sri Lanka is officially nonaligned, its government is firmly in the western camp. As the Army has become less able to control the insurgency, the government has stepped up its efforts to look for outside assistance from a wider spectrum of western countries, including the United States. The Trincomalee port is the best in the strategic Indian Ocean, and the conflict, already a regional problem, threatens to draw in outside participants.

For 13 years, this nation off the southeast tip of India has been plagued by a guerrilla conflict between the separatist Tamils, the predominantly Hindu group in the north, and the majority Sinhalese in the south, most of whom are Buddhists. A century of deeply rooted hatred, based on differences of language and religion, divides the two groups.

Since the summer of 1983, when Tamil guerrillas ambushed 13 Sinhalese soldiers in Jaffna, the latest wave of violence has spread increasingly southward, and much of the normal government infrastructure has now broken down. Basic services are still being provided, but the Sinhalese-controlled government is almost at a standstill, according to interviews with Jaffna residents by telephone and here in Colombo. Courts have not functioned for weeks, and all schools near the coastline have been closed because of a government-imposed security zone.

With the collapse of reconciliation talks in December, Sri Lankan officials appear to be turning increasingly to a military solution to the insurgency.

"We've underestimated the will of the Tamil people. Their dedication toward a state of Eelam is total. By the end of the year it will be one cry for separation," said a senior official who agreed to talk on the condition that he not be identified. Eelam is the name given to the separate state envisioned by Tamils.

The Jaffna Peninsula has been declared off-limits to foreigners, and repeated requests by correspondents for travel permits have been turned down.

Guerrilla commanders and fighters interviewed in the southern Indian state of Tamil Nadu, whose 45 million Tamils share their language and ancestry with Sri Lanka's 2.6 million Tamils, conceded in interviews that the supply line was crucial to sustaining the insurgency. They said the blockade has done little more than inconvenience the skippers of the fishing vessels and small motorboats that regularly make the two-hour run across the strait.

The Indian government repeatedly has denied allegations by the Sri Lankan government that it condones and even supports Tamil guerrilla training bases in southern India. It says that it merely has given refuge to approximately 40,000 Sri Lankan Tamils who have fled the fighting in the Northern Province. It is pressing the Sinhalese majority government of Sri Lankan President Junius R. Jayewardene for a political settlement to the Tamils' longstanding demands for autonomy and equitable representation.

The Sinhalese make up 74 percent of Sri Lanka's approximately 15 million people and the Tamils account for 13 percent. The rest are mostly Moslems and Christians.

Although separatist leaders in Madras claimed to have about 10,000 trained fighters either in Sri Lanka or moving constantly between Jaffna and southern India, Indian intelligence officials and western diplomatic sources say that the actual figure is closer to 2,000 and that many of them are not able to fight because of a shortage of arms.

In an interview at the official guest house in India's southern province of Tamil Nadu, separatist leader Appapillai Amirthalingam spoke openly about Tamils "reaching a point of no return" because of a Sri Lankan government decision to move 30,000 armed and trained Sinhalese settlers from the south to the Northern Province, where, since the escalation of rebel activity, most Sinhalese have fled.

Likening the plan to the Israeli settlement of the West Bank, Amirthalingam said, "If this colonization scheme goes through, there can be no solution other than the division of the country." He and other Tamil activists based in Sri Lanka warned that the settlement plan could lead to further clashes such as one Nov. 30, when Tamil guerrillas attacked a pilot settlement in the Northern Province and massacred 72 Sinhalese, mostly ex-convicts and their families, who had been moved there eight months earlier.

Buddhi Gunapunga, secretary of the Ministry of State and the Sri Lankan government's principal spokesman, said that the settlers, who will be given land and other financial incentives, are intended to achieve in the Northern and Eastern provinces the same demographic proportion as elsewhere in the country "so that in the future there won't be such a thing as a concept of a traditional ethnic homeland."

He acknowledged that to accomplish that goal in Jaffna, where more than 700,000 Tamils live and virtually no Sinhalese, or in Mullaittivu, where there are 60,000 Tamils and fewer than 1,000 Sinhalese, many times more than 30,000 settlers would be required. "The figures haven't been worked out yet," he added.

Since July 1983, when the 13 Sinhalese soldiers were killed in Jaffna, touching off anti-Tamil riots in the south that left more than 400 persons dead, 139 members of the security forces have been killed in the Northern and Eastern provinces and 382 civilians, mostly Sinhalese, have been killed by Tamil guerrillas, government officials said.

Sporadic violence recently has spread much farther south, to the heavily Tamil fishing town of Batticaloa on the island's east coast, where an ambulance was blown up by a land mine last week and four persons were killed.

R. Balasubramaniyam, secretary of the Jaffna Citizens Committee, said in an interview in Colombo that at least 2,000 Tamil civilians had been killed in what he termed "indiscriminate, unprovoked reprisal killings," and at least 4,000 arrested, 800 of whom are still being detained.

He added that creation by the security forces of a prohibited zone, or no man's land, 100 yards inland from the shoreline around the Northern Province into which no Tamils are allowed to enter, had put tens of thousands of fishermen out of work, resulting in food shortages in Jaffna.

Also, he said, a ban on unapproved travel in a "security zone" stretching from Jaffna town to Elephant Pass, the sole access to the Jaffna Peninsula, had resulted in fuel shortages.

"The situation is desperate in Jaffna, and all the security forces are doing is to drive the population to think more of separatism," Balasubramaniyam said. He said the Tamil guerrillas are in such control of the peninsula that Army troops venture out of their garrisons only occasionally in armed convoys.

One senior Sri Lankan official confirmed the Army's limitations, quoting Brig. Gen. Nalin Seneviratne, the northern commander, as saying, "Our authority is limited to the frontiers of our camps."

The government's military strategy is concentrated on cutting the supply line between southern India and Jaffna. Sri Lankan officials said they have ordered from a firm in Singapore U.S.-made Bell Ranger civilian helicopters, which they will modify for military uses by arming them with machine guns. They are also attempting to buy armored assault helicopters from the United States and other countries, but have not purchased any yet. They are negotiating with Britain for patrol boats, but for now, are buying new Chinese-made patrol boats to bolster the blockade.

They are also attempting to bring pressure on the government of Indian Prime Minister Rajiv Gandhi to isolate the guerrillas in Tamil Nadu.

Lalith Athulathmudali, the minister for national security, completed a trip to Washington late last month in which, officials here said, he urged Secretary of State George P. Shultz and other administration officials to try to persuade Gandhi that India's tacit support for the insurgency will only lead to destabilization of the Indian Ocean region.

Athulathmudali, whom Jayewardene gave wide powers last year in combating the insurgency, is understood to have conveyed to U.S. officials the same message that Prime Minister R. Premadasa gave last month to the Sri Lankan Parliament: "What would be the position if the Sikhs came to Sri Lanka to train in terrorism and fight for secession?"

Gandhi, however, is facing the same dilemma that his mother, Indira Gandhi had to contend with: To clamp down on separatist activity in Tamil Nadu would generate a strong reaction in an ethnically conscious and politically important state that the Congress (I) Party and its allies carried in the last parliamentary elections.

Moreover, as some Tamil leaders here and in southern India argue, if India dropped out of the equation, Sri Lanka could become a pawn in a superpower struggle because its Trincomalee port is the best in the stategic Indian Ocean.

"As long as both the United States and the Soviets recognize India as the mediator, there's no threat of this becoming globalized. But if there is any attempt to bypass India and one of the two powers interferes, it could become explosive. If superpowers get involved, we become like Lebanon," said Amirthalingam of the Tamil United Liberation Front.

Western diplomatic sources here and in New Delhi said that, given India's longstanding perception of itself as the preeminent regional power, it is not likely that Gandhi would end Indian involvement in the issue.

Some Sri Lankan officials, despairing of hopes for a negotiated settlement of the insurgents' demands, noted that even with only partial success in stemming the supply line to the guerrillas, the Army has forced the guerrillas into using more hit-and-run tactics rather than a sustained offensive. They are urging Jayewardene to concentrate on cutting that line.

"The terrorists say they can rout the Army out of Jaffna in three months. I believe that, but only if they can have an uninterrupted supply line," said a Sri Lankan government official. "India is the key".