In a meeting with Democratic and Republican congressional leaders, Mr. Bush said the $7.25 billion legislation "will bring some comfort" to unemployed workers.
The House overwhelmingly approved the measure earlier Wednesday by a vote of 416-to-4. The Senate passed the bill Tuesday on a voice vote.
The bill extends a federal program that provides 13 weeks of benefits for the unemployed who have exhausted their 26 weeks of state aid. The federal program lapsed on Dec. 28, but the Labor Department said the flow of benefits would continue uninterrupted if legislation were signed into law by Thursday.
Officials said an estimated 750,000 people are immediately affected, plus an additional 1.6 million who are expected to become eligible before the extension expires on June 1.
The lopsided votes masked a politically charged debate in which Republicans claimed credit for helping the jobless as their first order of business in the new Congress, and Democrats accused the GOP majority of acting grudgingly.
"It is important to note that the first piece of important legislation ... helps American families by extending unemployment insurance," said Speaker Dennis Hastert, R-Ill.
Still, Republicans fought making the program longer than 13 weeks, reports CBS News Correspondent Sandra Hughes.
"Nothing is good enough for them," said House Majority Leader Tom DeLay, R-Texas, saying Democrats would extend benefits indefinitely "so someone could stay out of work for the rest of their lives."
Democrats wanted to double the time workers would receive benefits.
"Not just 13 weeks, we need 26 weeks at least," said Democratic Rep. David Scott, a first-term Georgian who was sworn into Congress on Tuesday.
Congress adjourned late last year without passing an extension of the federal program, triggering angry protests from Democrats at the time.
At the same time, Rep. Bill Thomas, the California Republican who chairs the House Ways and Means Committee, said the debate was a precursor to a struggle over Mr. Bush's proposals to stimulate the economy.
Democrats have already criticized the proposals as heavily tilted toward the rich.
Speaking of Democrats who opposed the jobless aid bill on the floor during the day, Thomas said, "I'm quite sure my colleagues will be opposed to the proposals to stimulate the economy as well. So those will be future battles."
But for now, he said, "We're in a position of having the president sign a bill tomorrow or not sign a bill tomorrow."
Democrats said their proposals — an additional 13 weeks of benefits on top of the current 13 — could easily be accepted by the Senate and signed into law by the president. The change would help the estimated one million unemployed who have already exhausted their extra 13 weeks of benefits. The bid to alter the legislation failed on a vote of 224-202.
"The Republican Party does not care about those million unemployed," contended Rep. Jim McDermott, D-Wash., "because you have the presidency, you have the Senate and you have the House."
In fact, the measure that was sent to Mr. Bush's desk was far more generous than Thomas and other GOP leaders were willing to approve late last year, and more than even the Democrats were proposing in November.
The House approved a five-week extension in federal benefits last fall, but only for the jobless in three high-unemployment states.
In the Senate, Hillary Clinton, D-N.Y., and Don Nickles, R-Okla., reached agreement on an alternative at the time that would have extended the expiring program in all 50 states, through March 31.
Thomas and other House GOP leaders spurned the measure, though, and Congress adjourned.
In the Senate, newly elected Majority Leader Bill Frist, R-Tenn., sought passage of a five-month extension as the first order of business when the Senate convened on Tuesday. But by then, Democrats in both houses, noting that unemployment has continued to rise, decided to press for additional benefits for jobless people who have already exhausted their 13 weeks of benefits.
"Frankly, it's the least they can do," said House Democratic Leader Nancy Pelosi of California. "But we can do better."