"I am in a sense reluctant," Mr. Foley said today, "but I was convinced that these persistent rumors which have gone for so long and have occupied the attention of news media both here and abroad indicate that the inquiry should be taken."

Saying that the investigation is "certainly not political," Mr. Foley added, "If it were political, we would have started this earlier and had certainly less than reluctance to undertake it."

Over the years, various accounts have accused George Bush, then the Republican Vice-Presidential candidate, and William J. Casey, the Reagan campaign chairman and later Director of Central Intelligence, of traveling to Europe to persuade Iranian representatives there to delay any release until after the election. The hostages had been held by Iranian militants at the American Embassy since November 1979.

The issue of a possible deal faded after the 1988 election, but heated up again when Gary Sick, a former Carter Administration official and expert on Iran, wrote an article on the Op-Ed Page of The New York Times on April 15, saying that there was some strong evidence to suggest that the Republicans and the Iranians conspired in an effort to delay the release and weaken Mr. Carter politically. Michel Sees a 'Charade'

Today Bob Michel, the House minority leader, immediately moved to make the Republican case against the inquiry. "I think we're just going to be wasting an awful lot of money on a charade here," he said. He went on to say, "People back home don't give two hoots about it, what happened 10 years ago. Jimmy Carter lost the election."

The Democratic leaders did not announce a timetable for the investigations, but Congressional officials said they would begin in earnest after Labor Day, when Congress returns from its summer vacation.

The House's inquiry will be conducted by a special group headed by Representative Lee H. Hamilton, an Indiana Democrat who is chairman of the House Foreign Affairs Subcommittee on Europe and the Middle East. Members of the group, both Republicans and Democrats, will be announced in September.