He also said he did not regret his decision to join Mr. Bush in supporting the war in Iraq.
"I believe that we will remain a staunch and steadfast ally in the fight against terrorism," he said at a White House news conference with Mr. Bush.
Blair, once enormously popular in his country, saw his popularity tumble largely over his alliance with the United States on Iraq.
The president praised Blair, calling him extremely effective as a leader and "dogged" when he gets on a subject. "I appreciate the fact that he can see beyond the horizon. And that's the kind of leadership the world needs," Mr. Bush said.
Asked by a British reporter if Blair was the right person for Mr. Bush to be dealing with now, given that he will leave office on June 27, the president said absolutely. "You're trying to do a tap dance on his political grave," he said.
Blair said he was proud to stand shoulder-to-shoulder beside the U.S. since the Sept. 11, 2001, terror attacks. "I admire him as a president and I regard him as a friend," the outgoing prime minister said.
Mr. Bush voiced optimism that he could reach a deal with Congress on a stalled $124.2 billion spending bill to help pay for U.S. troops in Iraq and Afghanistan.
Congress and the White House have been at loggerheads over war spending since earlier this month when Mr. Bush vetoed the measure after the Democratic-controlled Congress added provisions for troop withdrawals to begin Oct. 1.
The president said he had instructed Joshua Bolten, his chief of staff, to stay in close touch with congressional leaders. He said he agrees with House Speaker Nancy Pelosi and Senate Majority Leader Harry Reid that time is of the essence, and that he respected the desire of members to include benchmarks in the bill that the Iraqi government should meet.
"I'm optimistic we can do so," Mr. Bush said.
Blair, noting that he could hear anti-war demonstrators outside the White House, defended anew his decision to join the U.S. and go to war in Iraq even though it has proven unpopular in both countries.
Even if people disagree with remaining in Iraq until victorious, "at least people understand that there is a battle we are fighting around the world today. ... You don't win those battles by being a fair-weather friend to your ally."
Blair had good words for Gordon Brown, Britain's Treasury chief, who was confirmed Thursday as the next leader of the Labour Party. Colleagues in the House of Commons overwhelmingly backed him as the only candidate to be the new prime minister.
"I wish him well. I believe he would make a great prime minister," Blair said.
Mr. Bush, however, acknowledged that he really didn't know Brown, although the two have met. "I hope to help him in office the way Tony Blair helped me," Mr. Bush said.
"Will I miss working with Tony Blair? You bet. Can I work with the next guy? Of course," Mr. Bush said.
The president was also asked about the possibility that World Bank President Paul Wolfowitz could lose his job over conflict-of-interest charges involving his girlfriend.
CBS News White House correspondent Mark Knoller reports it sounded as though Mr. Bush had accepted the inevitability that Paul Wolfowitz will have to step down.
"I regret that it has come to this,'' said Mr. Bush, who picked Wolfowitz for the job. "I admire Paul Wolfowitz ... I admire his focus on helping the poor."
He said Wolfowitz has the bank's best interests at heart.