Mr Lieberman is one of the Democratic Party's grandees, a vice-presidential candidate in 2000 who, two years later, ran for President. Today, however, he is in the fight of his life; a senator of 18 years standing who must endure the ignominy of a primary against a dangerous challenger who has built his campaign on his opposition to the war in Iraq.
The candidate himself remembers his brush with Bush slightly differently. "I don't think he kissed me," he told Time magazine. "He leaned over, gave me a hug, and said, 'Thank you for being a patriotic American.'"
But in anti-Bush and anti-war Connecticut, the dispute is academic. Bush's alleged words only remind voters of Mr Leiberman's still unwavering support for the invasion of 2003.
"He's a Republican mole in the Democratic party," says Pravil Banker, director of a financial company and a man who in other circumstances might be a natural Lieberman supporter.
"He's the guy who goes on [Republican-supporting] Fox News. He's the tame Democrat that even conservatives can stomach.
"He's too much a part of the Washington establishment. We must have someone who's in touch with us."
The setting for those forthright words is a an Indian restaurant in downtown Stamford, a booming town in the richest county of super affluent Connecticut, where representatives of the state's 50,000-strong Indian-origin community are hosting a 'get-to-know-you' event for Mr Lieberman's potential nemesis, a wealthy cable network executive called Ned Lamont.
As a primary opponent, Mr Lamont is surely an incumbent's worst nightmare. He is articulate and charming, a boyish 52-year old who graduated from Harvard and built an impressive business career, yet who finds time for pro-bono work at local schools.
His great grandfather was chairman of JP Morgan, bluest of Wall Street blueblood firms. His father served in the Nixon administration.
Only last year, Mr Lamont was a Lieberman contributor. He has switched from friend into foe - but in sorrow rather than anger. "I kept thinking Joe was going to change on Iraq, that he'd stand up and say it isn't right," he says. "But he did the opposite, and that inspired me to get into the race."
On Iraq, he says: "We've distanced ourselves from our traditions, we've ignored our values and are the weaker for it." But he rejects the Lieberman camp's charges that he's a single-issue candidate. "Iraq is only a part of it, there's a whole range of areas where the country has made bad decisions."
Among them are jobs, energy policy, and the country's failure to provide 47 million Americans - one in six of the population - with health coverage. "We have to insist on universal health care, and not go on spending $250m a day in Iraq."
And the message is getting through. At the start of the year, Mr Lamont trailed his opponent by more than 30 per cent. According to one recent poll, the gap is now down to 6 per cent among likely voters. And in the process, a local primary election has suddenly become a bellwether of the times.
The Lamont-Lieberman struggle is a battlefield in the civil war within his party. The race will set the tone for the mid-term election campaign this autumn, and have a large bearing on the contest for the Democratic nomination for the White House in 2008.
If all that sounds oddly familiar, it is. In 2004, a similar conflict played out, as the populist Howard Dean, a previously little known governor of Vermont became the darling of the activists. Propelled by internet-raised millions and the enthusiasm of his volunteer supporters from across the country, the governor briefly seemed a sure thing for his party's nomination.
Moderates were horrified. A vote for Mr Dean, warned Mr Lieberman, would be "a ticket for nowhere", that could sent the party "back to the political wilderness for a long time".
Today the roles are reversed. The bloggers and activists have rallied behind Mr Lamont, making him the symbol of what Markos Moulitsas, founder of the Daily Kos, current king of the liberal blogs, calls "the people-powered movement".
Daily Kos, and organisations like Moveon.org, have thrown all their energies into toppling Mr Lieberman. The primary outcome will thus also be a measure of the true influence of the blogs, held by some to be the new arbiters of American politics. Alas however, even the vote count on the evening of 8 August may not settle things.
In an extraordinary admission of weakness, Mr Lieberman has said he will run as an independent in the November general election if he loses the primary, and is already collecting signatures to that end.
Even he does lose, the likelihood is still that the next Senator of Connecticut will be named Joseph Lieberman. As a centrist, he is well placed to scoop up independent, even moderate Republican voters. One poll gives him 44 per cent in a three way general election, as much as Lamont and the official Republican candidate combined.
But in other ways, the damage is already done. The primary has rubbed raw Democratic divisions over Iraq, offering a precious opening for Republican master strategist Karl Rove. Once again the man called 'Bush's Brain' is casting the upcoming midterm elections as a referendum on national security, traditionally the weak point of the Democrats. Once again charges are flying that Democrats will just "cut and run" from Iraq and the rest. And if the 'kiss of death' helps that strategy, then from Mr RoveÕs point of view, so much the better.
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