The Reagan administration began its final push yesterday to yield control of National and Dulles International airports to a regional commission, but even its most committed supporters say the idea faces an uphill struggle in the House of Representatives.

Flanked by a wide array of major Virginia political figures, U.S. Transportation Secretary Elizabeth Hanford Dole went to the steps of the Capitol to call for passage of the transfer legislation.

Even as Dole spoke, Rep. Norman Y. Mineta (D-Calif.), chairman of the House aviation subcommittee, introduced a bill calling for a federal corporation to operate the two airports. They are currently run by the Federal Aviation Administration, the only commercial airports in the country operated by the federal government.

"That isn't good enough," Dole said when asked about Mineta's bill. "The federal government should not be in the business of operating airports," she said. "Period."

The Transportation Department has estimated that more than $700 million is needed for renovations at National and expansion at Dulles, funds that the administration said it cannot afford to spend. Supporters of the regional commission said it can easily raise the money for needed improvements.

"National Airport has become a national disgrace," said Virginia Gov. Gerald L. Baliles at the Capitol news conference. "They used to sell life insurance before you got on the plane; now they sell it to you when you get out of your car.

"The truth is, National is a joke without a punch line, a comedy without the laughter."

The purpose of Mineta's bill is to allow the government to retain control of the airports while financing renovations there through bonds issued in the name of the federal corporation. Transportation Department and congressional sources said that Mineta introduced the bill, contending that there is not sufficient support in the House to pass the legislation proposed by the administration.

It has been two years since Dole first urged the creation of an authority to run the airports. Supporters are growing increasingly concerned that, like past plans, this one is headed for oblivion.

"This is so eminently sensible," Dole said yesterday, reminding her listeners that there have been eight previous and unsuccessful attempts to remove the two airports from federal ownership. "If we don't do it now, when will we do it?"

In April, the Senate passed a measure that would permit the airports to be leased to a regional authority that could then raise revenue by floating revenue bonds. The House has shown less interest in ridding the government of the airports, which provide coveted free parking spaces for members of Congress.

In the Senate, the Reagan administration overcame a lengthy filibuster by Paul S. Sarbanes (D-Md.) after vigorous lobbying by Secretary Dole. Maryland officials contend that a regional authority would create an unfair competitive advantage for Dulles over Baltimore-Washington International Airport, which is owned by their state.

Some Maryland officials, particularly those from Montgomery County, have objected to the Senate version of the bill, saying that it would allow the new authority to discard strict noise standards that are now in force at National.

There are other obstacles in the House.

The most intractable may be an amendment to the Senate bill that would allow nonstop flights from National to go 1,250 miles, an increase over the current limit of 1,000. The limit is known as the perimeter rule, and dispensing with it would allow direct service from National to Dallas-Fort Worth and Houston airports.

The perimeter rule was originally designed to help Dulles, which for decades was badly underused.

Some House members from smaller cities have expressed concern that an increase in long-distance flights from National would mean fewer direct flights to their region. Rep. Gene Snyder (R-Ky.), the ranking minority member on the House Public Works and Transportation Committee, long has fought any expanded perimeter for National.

Any debate of the rule on the House floor is expected to provoke a struggle between Snyder and Majority Leader James C. Wright (D-Tex.). Wright has long advocated expanding the perimeter so that his constituents can fly nonstop to National Airport.

Another problem faced by the administration bill is that its principal House sponsor, Rep. Frank R. Wolf (R-Va.), is up against the toughest opposition he has seen in recent years this fall. He will run against Arlington County Board Member John G. Milliken, and the House Democratic leadership is not eager to hand Wolf a big major campaign asset.

"We have some real problems, I guess," said one Transportation Department official who asked not to be named. "But nobody ever thought we would even get this far."

Baliles put it another way.

"Our message is drawn from the Old Testament," he said yesterday. "Let our airports go."