WASHINGTON  The House Republican leader, John A. Boehner, on Sunday opened the door to a compromise on the contentious issue of the Bush-era tax cuts, saying he would vote to maintain lower rates for families earning less than $250,000 even if President Obama and Democrats insisted on ending the cuts for wealthier Americans.

With Congress returning to Washington this week, Mr. Boehner’s decision is likely to reframe both the final intense weeks on Capitol Hill before the elections and the fall campaign, in which embattled Democrats have planned to paint Republicans as obstructionists favoring the rich over the middle class.

Mr. Obama in recent days has lambasted Congressional Republicans  at times singling out Mr. Boehner by name  for threatening to block an extension of the tax cuts for more than 97 percent of Americans to preserve lower rates for “millionaires.” And Democratic party leaders had been planning to use the tax fight to help rally their base.

“If the only option I have is to vote for some of those tax reductions, I’ll vote for them,” Mr. Boehner said in an interview on “Face the Nation” on CBS, during which the host, Bob Schieffer, pointedly asked whether Republicans would hold the tax breaks for most Americans “hostage” to keep the lower rates for the wealthy.