The likely GOP nominee sought to send a message that he recognizes the close bonds between the U.S. and its top ally -- and to project an image of leadership.
"We have a very special relationship between the United States and Great Britain," Romney told NBC News in an interview in London on the first day of a weeklong overseas trip that will also take him to Israel and Poland.
"It goes back to our very beginnings -- cultural and historical."
Romney's first official appearance during a campaign swing intended to highlight longtime U.S. alliances was with Blair.
He was slated to meet later in the day with current Prime Minister David Cameron.
Blair hosted Romney at his private office few blocks off Hyde Park.
The former Labour Party prime minister now serves as a special envoy to the Middle East for the British government.
The two discussed the Olympics and exchanged pleasantries at the beginning of a planned half-hour meeting.
Romney also is slated to meet with Nick Clegg, the deputy prime minister, and Ed Miliband, the current leader of the Labour Party.
Romney, a former private equity executive, also requested a meeting with Chancellor of the Exchequer George Osborne, Britain's top financial official.
Accompanying Romney to some of his meetings are former Missouri Sen.
Jim Talent, an adviser, and Kerry Healey, who served as lieutenant governor when Romney was governor of Massachusetts.
Romney, whose decades in private business gave him ample exposure to international affairs, is a former one-term governor untested on the world's political stage.
He hopes to convince voters back home that he is no novice on foreign affairs and that they should elect him as president in a complex, dangerous world.
Romney also will spend part of his time in London raising money and highlighting a key part of his resume -- the successful Salt Lake City Olympics he managed -- with an appearance Friday at the opening ceremonies of the London Games.
Meeting with British officials is typically one of the first priorities of any new president, and establishing those relationships beforehand can help smooth any transition.
It's not unusual for American presidential candidates to meet with British leaders during the campaign; Obama did so when he took a trip abroad as the likely Democratic nominee in 2008.
This isn't Romney's first meeting with Cameron; the two also talked during a Romney visit to London in 2011.
This year, Cameron traveled to the U.S., where he met Obama and attended a state dinner in Washington but did not meet with Romney.
Romney's meeting with a deputy prime minister is somewhat unusual. It's happening because Britain has a coalition government, and Clegg's Liberal Democrats govern alongside Cameron's Conservative Party.
The meetings come a day after the Daily Telegraph newspaper published a story quoting an unidentified Romney campaign adviser saying the Republican believes the U.S. relationship with Britain is special because of shared "Anglo-Saxon heritage" and the White House doesn't appreciate that shared history.
Romney, however, quickly distanced himself from any such view.
"I don't agree with whoever that adviser might be," Romney told NBC News, "but do agree that we have a very common bond between ourselves and Great Britain."
Nonetheless, Vice President Joe Biden and top Obama aides criticized Romney. "The comments reported this morning are a disturbing start to a trip designed to demonstrate Gov. Romney's readiness to represent the United States on the world's stage," Biden said.
Later Thursday, Romney planned to hold a high-dollar fundraiser at the swanky Mandarin Oriental hotel in London's tony Knightsbridge district.
One of the hosts of that fundraiser, former Barclays CEO Ian Diamond, withdrew from the event after he resigned in the wake of a rate-rigging scandal wracking British banks.
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