Dana Perino, a Bush spokeswoman, said: "The president has said that we need to get the process over with, in terms of them sending him a bill and him vetoing it so that we can take the next step.
The move will be his first veto confrontation with Democrats since they took control of Congress in January.
Democratic Senator Robert Byrd of West Virginia said: "The troops had the courage and the strength to win the war, but the president has not had the wisdom to win the peace."
Democrats, however, are unlikely to be able to defy Bush because they do not think they have the two-thirds support in Congress required to overturn a presidential veto.
Republicans said the vote amounted to little more than political theatre because the bill would be dead after it reached the White House.
Unlike an earlier version the House of Representatives passed last month, this bill would not set a firm date for all US combat troops to leave the war.
Instead, a non-binding March 31 date for finishing the withdrawal merely would be a "target".
The bill would allow some US troops to stay in Iraq beyond March to continue training Iraqi soldiers, protect US facilities and to conduct targeted counter-terrorism missions.
Mitch McConnell, the Senate Republican leader, said: "The solution is simple: Take out the surrender date, take out the pork, and get the funds to our troops."
Democratic leaders could draft a new bill giving money to the troops in Iraq, possibly with some watered-down conditions Bush could accept, and leave the fight over troop withdrawals for future legislation.
The US has recently sent over 20,000 extra troops to Iraq as part of a crackdown aimed at ending the brutal sectarian violence that has gripped the country.
Speaking at the Pentagon on Thursday General David Petraeus, the US commander in Iraq, said the war effort probably will "get harder before it gets easier".
Petraeus said the situation in Iraq was "excessively complex and very tough".
He said there have been some improvements in the two months since Bush's troop build-up began, but "there is vastly more work to be done across the board... We are just getting started with the new effort."