The Federal Aviation Administration yesterday proposed elimination of a 14-year-old rule restricting the number of flights daily at three of the nation's busiest airports.

The so-called high-density rule applies to LaGuardia and John F. Kennedy in New York and O'Hare in Chicago. Washington National Airport is also covered by that rule, but its flights would continue to be restricted by the FAA even if the rule is dropped.

The proposal to eliminate the rule comes in response to a 1981 petition from United Airlines, which wants to bring more flights than currently permitted into its major hub at Chicago.

"In a perfect world, we would retime some flights and bring more in at the same time so passengers would have more convenient selections" when changing planes, said United spokesman Joe Hopkins. Other airlines contacted also applauded the proposal.

As a practical matter, flight restrictions imposed after the strike and firing of 11,400 air traffic controllers in 1981 will have to remain for some time to come, the FAA said in the Federal Register.

But then, new air traffic procedures and increased computer capacity will allow more flights to use the three big airports than are permitted under the high-density rule, the FAA said.

The New York Port Authority, which operates both LaGuardia and Kennedy, opposed United's petition, saying that congestion created by hundreds of passengers arriving and departing from jumbo jets, particularly at LaGuardia, was reason enough to restrict flights.

The FAA's proposal said, however, that "ground congestion is not a basis for retaining the high-density rule."

Robert Aaronson, now director of aviation for the port authority and FAA airport policy chief during the Carter administration, said, "We're very concerned about the basic issue of giving airlines the feeling there are no limits to the number of airplanes they can shoehorn into an airport."

The FAA plans two hearings on its proposal and will hold the record open for comments on it until June 3.