What links the speculation in Washington over the next president of the World Bank with an avidly awaited blockbuster documentary that kicks off tomorrow night on television here? Or, to put the question another way, which is the most mesmerising family in American politics?

The Kennedys, you might say, given the fuss over the recent book by the former White House intern Mimi Alford about her affair with JFK. Or you might pick the Obamas, the current residents of 1600 Pennsylvania Avenue, or the Bushes, a three-generational dynasty rivalled in modern times not even by the Kennedys, featuring a senator, two governors and two presidents. But no. In terms of their grip on the public imagination, their star power and their enduring relevance, none quite match Bill and Hillary Clinton.

Right now, the spotlight is on Hillary. She is universally considered to have been an excellent Secretary of State, more admired by the public than any other member of the Obama administration or Congress. That she plans to leave her current post by the end of Barack Obama's first term at the latest is well known – but what then?

A presidential run in 2012 is out of the question, and so surely is one in 2016, when she will be 69. But some Democrats still fantasise about a job swap with Joe Biden this summer, to produce an Obama-Clinton dream ticket for November. Now a new possibility has opened up. When Bob Zoellick formally announced last week he would step down in June as head of the World Bank, at the top of the list of likely replacements was Hillary.

From the interested parties, the denials have come thick and fast, but not entirely convincingly. She herself has often said that four years at the State Department is enough, and has evinced no visible interest in becoming vice-president. The World Bank, however, would be less demanding, yet cater to her keen interest in global development and human rights, especially women's rights.

In a month or two we'll have the answer. But Bill's return to prime time is already upon us. Tomorrow, as every third Monday in February, Americans celebrate Presidents' Day – and this year PBS is marking the occasion with the first episode of Clinton, a four-hour examination of the life of the 42nd president, on its flagship documentary series American Experience. At the very least, a cynic might note, fans of upmarket soap opera will be spared withdrawal pangs. Tonight PBS is running the finale of the second season of Downton Abbey, whose depiction of the British aristocracy during the reign of George V seems to have transfixed every sentient American. But no problem. Enter the star of America's real-life political soap, at the opposite end of the 20th century.

Bill Clinton, of course, was no aristocrat. He was brought up in the shadow of an abusive and alcoholic stepfather in redneck Arkansas. Unlike the earls and countesses of Grantham, he inherited nothing. From his university years, he wanted to be president – and he achieved that goal thanks to his own charm, brains and ambition, and his prodigious gift for politics.

But a decent soap also needs plenty of plot twists and plenty of flesh, and Bill Clinton's life provides both in abundance. Ask anyone to sum up his presidency in two words, and all but certainly they would be "Monica Lewinsky", the White House intern with whom he conducted his own squalid liaison, risking his job, his marriage and his reputation in history. Part one of Clinton starts with a flash forward to the Lewinsky affair, while the second episode is largely given over to it. But with Bill no reverse was ever permanent. His ability to get himself into trouble was exceeded only by his capacity for getting out of trouble – almost as if he were bored by a "normal" super-successful existence. The real thrill lay in getting out of impossible jams, and so he recklessly created such jams.

Republicans were driven crazy by their inability to nail him. All but the last two of Bill Clinton's eight years in office were studded by scandals real and imagined, from the impenetrable Whitewater affair to a succession of "-gates". And then there were the "bimbos", whose favours he sought and secured only too often. Bill must be the most investigated president of modern times. Even so, he left office with the highest approval rating of any president since the Second World War.

In a sense, American Experience consigns him to history; indeed, the Clinton era, with its prosperity and balanced budgets, in the interlude between the collapse of Communism and 9/11, already feels like a vanished age, almost as remote as Downton Abbey. But the man himself is very much with us, an elder statesman of only 65, his past follies long forgiven as he travels the world, heading a philanthropic foundation marshalling hundreds of millions of dollars but whose greatest asset is his own charisma.

Bill's interests are truly global. Last year he produced Back to Work, a manifesto on reviving the US economy. In the past week alone he has chaired an "Invest in Ireland" meeting in New York, held forth on the violence in Nigeria and attended a pro-am golf tournament in Colombia. And, like Hillary, he's also been mentioned for the World Bank job.

Constitutionally, he is barred from a third term – and if his wife ever did make another presidential run, would Americans really want another Clinton psychodrama in the White House, even with the roles reversed? Well, maybe they would. Through all the ups and downs, all the scandals, the Clinton marriage has survived, its partners as mesmerising as ever.