A Reagan administration bill to shift control of National and Dulles International airports from the federal government to a regional authority has run into fierce and unexpected opposition on Capitol Hill.

The combined forces of Maryland politicians who say the transfer would be unfair to competing Baltimore-Washington International Airport and of fiscal conservatives who are taking a tough new look at the federal budget have put the bill in jeopardy.

Yesterday, Sen. Gordon J. Humphrey (R-N.H.), a conservative whom the administration had counted on for support, announced in a news conference that he thought the government ought to look at other options, including the possibility of selling the airports to the highest bidder.

"In the face of budget cuts and a $180 billion deficit, it seems to me we ought to consider selling these airports at fair market value," he said. "We have a private party that has expressed interest in buying them. We owe American taxpayers at least an examination of that potential offer."

Last week, a British bank, N.M. Rothschild and Sons, said that if the U.S. government indicated a serious interest, it would seek to raise up to $1 billion to back a private group that would purchase the airports.

But yesterday, U.S. Transportation Secretary Elizabeth Hanford Dole, who has been the principal force behind the transfer bill, said the government would oppose any effort to sell the airports.

"The idea of selling the airports to British investors or anyone else at a high price is completely unrealistic," she said. "To pay off such an investment, airport fees and charges would be passed on directly to airport users in the form of higher ticket prices. Congress would not tolerate such high-cost airports."

For four days, Sen. Paul S. Sarbanes (D-Md.) has filibustered to block debate on the proposal, which administration officials had expected to pass easily in the Senate. Senate backers could not muster the 60 votes necessary Friday to cut short the filibuster, and administration officials and Senate staff members said they expect that a second attempt today will be close.

"We are feeling very frustrated by the whole experience," said a Transportation Department official who has been working on the transfer legislation and who asked not to be identified. "We expected nothing like this in the Senate."

Representatives of the group that said it was willing to bid "$500 million to start" for the two airports said at yesterday's news conference that it would improve parking and terminal facilities immediately if it obtained National and Dulles.

Officials say both airports need massive infusions of capital, and the federal government, which owns and operates them, has invested almost nothing in their development in the last few years. Airline officials and development specialists have said that National and Dulles will need up to a total of $500 million worth of investments in the next five to seven years.

The British bank became interested in the purchase of the airports last month when John Redwood, director of international operations involving the takeover of public facilities, attended a conference on such takeovers that was sponsored by the U.S. Agency for International Development.

Representatives of major U.S. airlines joined the Transportation Department in attacking the private purchase proposal yesterday. Airlines long have sought to keep airports nonprofit, fearing that private owners would raise landing fees.

"We hate this proposal, and it is without merit," said Clark H. Onstad, vice president of Texas Air Corp., which has a large number of daily flights in and out of National.

He said that selling the airports at market value would cause enormous increases in fees for landing rights at the two airports and that the added costs would be immediately passed on to passengers. A landing at National during peak hours costs $184; Onstad said that if National were sold to the highest bidder, the price could increase ninefold.

Even if the Senate gives no further consideration to a sale of the airports to the highest bidder, the proposals -- and the delay in Senate passage -- have cost the bill support.

Supporters of the transfer to a regional authority had counted on easy sailing in the Senate to persuade members of the House to pass the legislation.

No hearings are scheduled in the House, but officials there have said they will look more closely at the bill if it passes the Senate.