Rep. Duncan Hunter of California, chairman of the Armed Services Committee, had been one of the main roadblocks to passage of a bill implementing the Sept. 11 commission's terror-fighting recommendations.
Hunter had wanted the bill to ensure that the Pentagon would retain some control over the tactical agencies that operate the nation's spy satellites and analyze the results for troops on the battlefield.
He said Monday in a joint statement with Senate Armed Services Chairman John Warner, R-Va., "Pending a review of the rest of the legislation, we are prepared to support the bill as amended by this new language."
President Bush invested some of the political capital he earned on Election Night to reach a compromise Hunter could live with, reports CBS News Chief White House Correspondent John Roberts.
"I'm very optimistic now that the bill will move forward and that we have the votes to win in both the House and the Senate," said Sen. Susan Collins, R-Maine, chairwoman of the Senate Conference Committee.
That would be a victory for many families of 9/11 victims, who kept vigil outside the White House on Monday.
"How many more people have to die before you're willing to make this country safe?" said one woman.
It's also a victory for the president, whose arm-twisting narrowly avoided an embarrassing post-election legislative defeat, one that could have had lasting impact on his second-term agenda.
"If he gets rolled here, on this bill, imagine what it will be like when he tries to come forward with other controversial proposals," said Rep. Jane Harmon, D-Calif.
House Speaker Dennis Hastert had refused to move the legislation to a vote before Thanksgiving because of objections from GOP chairmen including Hunter and House Judiciary chief James Sensenbrenner of Wisconsin.
Sensenbrenner remains opposed to the bill because he wants such issues as illegal immigration and asylum changes dealt with as well.
With an agreement in place, House Republicans probably will meet Tuesday morning to decide how to move the bill forward.
At the White House, spokesman Scott McClellan said, "We feel very hopeful that this legislation will get passed this week."
Earlier Monday, Mr. Bush said, "I believe we've addressed the concerns, by far, of the majority of the members of both the House and the Senate."
Mr. Bush, speaking during an Oval Office meeting with Iraq's interim President Ghazi al-Yawer, said, "It's a piece of legislation that is important for the security of our country."