U.S. and French troops of the multinational peace-keeping force here came under artillery fire from the mountains east of Beirut again today despite warnings of possible strong military countermeasures to end the repeated attacks.

Meanwhile, heavy fighting continued for the sixth day between Christian and Moslem Druze militia forces battling for control of strategic positions in the mountains, and by nightfall the Druze claimed to have routed the Christians from two more towns and to have surrounded a third where thousands of refugees have gathered.

The United States was reported to be "accelerating" the shipment of ammunition and other supplies to the Lebanese Army, which is also fighting to keep the Druze and their Lebanese and Palestinian leftist allies from gaining access to the capital.

On the diplomatic front, U.S., Saudi and Lebanese efforts to arrange a halt in the fighting intensifed today, and a Lebanese spokesman predicted that "the situation will crystallize very soon." But another government bid to get at least a partial cease-fire was reported to have failed.

[The State Department called for an immediate cease-fire in Lebanon, saying that further bloodshed cannot serve the cause of peace and national harmony. Deputy spokesman Alan Romberg said, "We appeal to all those involved in the fighting to respect human life and alleviate the suffering that war has brought. Nothing good can come of indiscriminate killing, regardless of its perpetrators or its victims."]

U.S. Marine positions at the Beirut international airport were hit early this morning by 12 rockets and then again in the mid-afternoon by 12 rounds of 81 mm mortar fire.

No casualties or major damage occurred, and the Marines did not respond to the fire as they did yesterday with ship and shore guns after a similar shelling.

The headquarters of the 2,000-man French contingent, located at the former ambassador's residence in Beirut, was also shelled early today and a French officer was injured. At dawn, two French Super Etendards flew over the city and mountains to the east on a reconnaissance flight but fired no weapons.

Both France and the United States have been sending strong signals to the Druze militiamen and their Syrian backers believed responsible for the shellings that the considerable sea and air power at the disposal of the multinational force will be used if the attacks do not cease. But so far the warnings have had no visible effect.

Yesterday, President Reagan assured Col. Timothy J. Geraghty, commander of the 1,200-man Marine contingent, by telephone, that the United States would provide "whatever support it takes to stop the attacks on your positions."

But the Marines already have an impressive amount of air and naval firepower standing off shore, including two aircraft carriers. The real question seems to be the political decision of how much use to make of them.

The British Defense Ministry sent six Buccaneer warplanes to Cyprus, 150 miles west of the Lebanese coast, to provide air support for the 100 British soldiers in the peacekeeping force, The Associated Press reported.

The shelling is believed to be the work of Druze militiamen who hold most of the high ground overlooking the airport and capital from their stronghold in the mountainous Chouf region southeast of the capital.

But Marine spokesmen here do not discount the possibility that either the Syrian Army or leftist Lebanese and Palestinian factions aiding the Druze may also be involved in the shelling and acting on their own to provoke a confrontation.

"I'm sure there are individuals and factions that are trying to test us and trying to get us to discredit our role here as neutral," Marine spokesman Maj. Robert Jordan said.

Last night Druze leader Walid Jumblatt told a press conference in Hamana behind Syrian lines that the four-nation, 5,400-man multinational force was "getting more and more one-sided" in favor of the Christian militia and the government of President Amin Gemayel.

"If this continues we shall demand the departure of the MNF to where it came from," he said. "We want no fleets in our waters. We want no new colonialism."

Meanwhile, in the mountain fighting today, Jumblatt's Druze militia continued to make progress in its campaign to drive the Christian Phalangist militia completely out of the Chouf. The Druze announced the capture of Bayt ad Din and recapture of the strategic road junction at Qabr Chamoun, which it had taken and then lost in heavy fighting yesterday and today.

The junction is crucial to both sides because it permits the Druze to link up their positions on the Beirut-to-Damascus highway to those in the southern outskirts of the capital.

A Phalangist spokesman in Beirut acknowledged the "evacuation" of Bayt ad Din but said it was done because "we don't have enough troops."

He denied reports circulating here that the Christian militia was on the point of evacuating the entire Chouf and he insisted it was still in control of Qabr Chamoun after beating back an early morning Druze offensive in which he said two Palestinian battalions had also participated.

But a Druze spokesman said later tonight that the Druze militia had retaken the town in the afternoon.

Meanwhile, attention here continued to focus on the plight of a reported 25,000 to 40,000 Christian refugees trapped in the town of Deir Qamar in the southern Chouf.

The Druze have surrounded and have been shelling the town, raising fears for the fate of the refugees, who reportedly are running short of food and water. The Christians fear a massacre if the Druze take the town and they have been pleading for international help to prevent killing.

Tonight, Jumblatt, in an interview with Radio Monte Carlo, sought to reassure the Christians.

"I give the Christian population of Deir Qamar and the refugees in that town my word that they will not be harmed," he said.

The International Red Cross attempted again today to send in a four-truck convoy with enough food to feed 30,000 people for three days. The convoy was supposed to be escorted by the Israeli Army from Jezzine, in the Israeli-controlled portion of southern Lebanon.

But the convoy reportedly was stopped by Druze when it reached Bayt ad Din. A Red Cross spokesman said tonight that the mission had run into some unexpected trouble and that the convoy had to return to Sidon. He did not elaborate.

Christian sources said Gemayel had proposed a deal last night to the Druze and Christian militia leaders whereby a cease-fire would be declared in the Deir Qamar and Qabr Chamoun areas allowing each side to hold on to its respective positions and the Red Cross to send in relief supplies to the refugees.

Christian militia spokesman Fadi Hayek said the Phalangists had accepted the deal but that it was upset after the Druze attack today on its forces around Qabr Chamoun.

He charged that the Druze intended to use the threat of a massacre of the Christian refugees in Deir Qamar to force the Phalangist forces to end their fight to stay in the mountains.

"Now they are using Deir Qamar as a hostage," he said. "Whenever we move, they will say, 'Better stop or we will kill the civilians.' "

On the diplomatic front, Saudi Arabia today sent its special envoy Prince Bandar to Cyprus to meet Gemayel's national security adviser Wadie Haddad to discuss the crisis.

Later, Haddad returned to Beirut where he met with Gemayel in the presence of U.S. special envoy Robert C. McFarlane to discuss the results of the talks while Bandar returned to Damascus for more discussions with Syrian leaders.

A Lebanese government spokesman here said contacts with all sides to arrange a cease-fire in the mountain war and discuss a larger overall political settlement were continuing and that the results should crystallize "very soon."

The elements of a compromise are said to include a cease-fire linked to the withdrawal of the Christian Phalangist militia from the Chouf and the entry of the national Army there; the formation of a national unity government composed of the country's main political leaders, and a conference to discuss a new power-sharing formula among Lebanon's main ethnic and religious groups.

It is not clear how much progress is being made in these mediation efforts, and many here feel that the fighting is likely to continue at least a few more days until the relative strength of the two warring militia forces in the Chouf becomes clearer.