The Iranian-based Al Dawa organization, which has been implicated in several hijackings, kidnapings, assassination attempts and suicide bombings against U.S. targets throughout the Middle East, appears to be widening its sphere of operation and gaining the recognition of some radical Arab countries.

During a recent trip to Syria and Libya by members of the Supreme Assembly of the Islamic Revolution of Iraq, of which Al Dawa is a member, Syrian authorities allowed the shadowy movement to open offices in Damascus while Libyan officials promised to coordinate with Al Dawa and allow it more freedom of action for "combating imperialism," Syrian and Iranian newspapers reported.

Al Dawa's expanding operations could take more significance in the light of the release in September of the Rev. Benjamin Weir, an American held captive in Lebanon for 16 months. His captors threatened to kidnap more Americans if Kuwait does not release 17 Shiite Moslem prisoners convicted in a series of fatal bombings in Kuwait in 1983.

According to Arab diplomats and their intermediaries in Washington, about half of those prisoners are Iraqi members of Al Dawa. Their release has been demanded in several terrorist incidents during the last 18 months.

In one such case, Arab diplomats in Washington say, Mofti Akeel, an Iraqi member of Al Dawa living in Lebanon, was sent to Kuwait to attempt to kill that country's ruler, Sheik Jabir Ahmed Sabah, last May.

After the unsuccessful attempt, an anonymous caller told news agencies in Beirut, "We hope his majesty and the Kuwaiti government have got the message. We want our 17 brothers who have been kept in jail there to be released."

Reliable sources close to the Iranian government say that two Al Dawa members were among the four men who hijacked a Kuwaiti passenger plane last December, forced it to fly to Tehran's Mehrabad airport and shot to death two Americans working for the Agency for International Development. The other two hijackers reportedly were Iranian citizens.

Iranian security forces arrested the hijackers and said they would be put on trial, but there has been no further information about such a trial.

Al Dawa, according to the sources in Washington, is not an independent group but works under an umbrella organization called the Supreme Assembly of the Islamic Revolution of Iraq.

That group, established in Tehran in September 1981, is widely believed by anti-Khomeini sources in Tehran, Washington and London to have control over two dozen Islamic fundamentalist and terrorist groups thorughout the Middle East, including Al Dawa.

The assembly is headed by an Iraqi-born Shiite fundamentalist, Mohammed Baqer Hakim, a relative of the Iraqi opposition leader Mohammed Baqer Sadr, who was arrested with a score of Al Dawa members and sympathizers in 1980 and executed. As leader of Al Dawa, sources say, Hakim coordinates the movement's activities in frequent trips to England, where his brother Mehdi Hakim has been granted political asylum and runs the Ahl Beit Islamic Society in Manchester.

Al Dawa appears to have ample financial resources, and is said by anti-Khomeini sources in London to have spent nearly $500,000 on advertisements against the Iraqi-regime and propagating Islamic fundamentalism in European papers, mostly in Britain, from 1981 to 1983. At least some of the money came from private donations and it is not known whether any came from the Iranian government.

A close associate of Mehdi Hakim in London is Mehdi Bahr Omum, an Iraqi clergyman closely allied with Iran who is known as "head of the militant Iraqis abroad."

In addition to his leadership of Al Dawa, Baqer Hakim, as the spokesman of the Supreme Assembly of the Islamic Revolution of Iraq, works closely with at least two other active Shiite fundamentalists, one of them a top-ranking Iranian official.

One of them, Hadi Modaressi, Ayatollah Ruhollah Khomeini's religious representative to Kuwait, heads the Islamic Front for the Liberation of Bahrain. He and his brother, Mohammed Taqi Modaressi, who formed the Islamic Action Party in 1979 with the aim of overthrowing Bahrain's government, have been allowed by Iran to broadcast a daily four-hour Arabic "Voice of Islamic Revolution in Bahrain" program beamed to Bahrain from the state-run Tehran radio.

The program recently called on the people of Bahrain, according to the London-based Al Dastor magazine, "to take to the streets and resist with your chests the bullets of the soldiers of the ruling regime in Bahrain and . . . learn the lessons of the revolution in Iran."

Hakim's other close associate is Ali Akbar Mohtashemi, Iran's ambassador to Syria, who acts as a liaison officer between the Iranian government, pro -- Iranian groups in Lebanon and the Syrian regime of Hafez Assad.

Washington Post staff writers Richard Harwood and Bob Woodward reported in 1984 that U.S. intelligence officials think Mohtashemi was the person through whom a "fixer" named Hassan Hamiz arranged for the car-bomb explosions at the headquarters of the U.S. Marines and French paratroopers in Beirut.

While Al Dawa has no official representation in the United States, a handful of other organizations opposed to the Iraqi regime operate actively here, under the sponsorship and financial support of pro-Khomeini groups, some with official links to the Iranian government.

One of the best-known is the Islamic Union of Iraqi Students, based in Lawrence, Kan. Tehran's semiofficial English-language newspaper Tehran Times reported on Sept. 10 that a seminar held by the union had sent a telegram to Iran's leaders, paying homage to the guidance of the Supreme Assembly of the Islamic Revolution of Iraq and to Khomeini's "universal leadership."

In a recent request filed with the D.C. police for a permit to hold a demonstration here, the group gave the telephone number of the Islamic Education Center in Potomac. The center is run and funded by pro-Khomeini elements, many of whom have official contacts with Iranian officials and the Iranian Interests Section of the Algerian Embassy here.