But late on Wednesday, they had to admit defeat, and in a move without precedent in their six years at the White House, the First Couple issued a pained statement of protest against a specific publication.
"Unfortunately, despite personal appeals with respect to her privacy and her security from her parents, People has chosen to run the story," it said.
"We deeply regret and are profoundly saddened" by the decision, the Clintons said.
People, owned by Time Warner, was unapologetic. Its managing editor, Carol Wallace, said: "Over the years we've written about the Clinton family in good times and bad. Chelsea is nearly 19 years old and a poised young adult. We feel that because she is an eyewitness to the family drama and historical events unfolding around her, she is a valid journalistic subject."
For Hillary, but especially for Bill Clinton, the end of the US media's respect for Chelsea's privacy may be the cruellest consequence of the Lewinsky affair. Chelsea is a second-year medical student at Stanford University in California and her closeness to both parents has long been evident, as has their concern to shield her from the limelight.
During the 1996 presidential campaign, they rejected her pleas to join the campaign trail, but have gradually loosened the leash, relying on what those familiar with the British tabloids would find to be the remarkable restraint of the US media.
Parental calls for her to be left alone when she went to college 18 months ago were largely heeded by the press, with only the odd diary note about a boyfriend.
Chelsea, meanwhile, regularly accompanied her parents on foreign trips, most recently to China. Last year, she also travelled with Hillary on her African tour, a trip that marked a debut of a kind into public life.
However, with the rumours, and finally the confirmation, of the President's affair with awoman only five years older than his daughter, the media's restraint started to break down. The poignant pictures of Chelsea, walking hand-in-hand between her parents as they crossed the White House lawn to the presidential helicopter the day after her father's confession, were taken as symbolic - Chelsea was holding the family together.
Such public appearances and her spontaneous move to gladhand the crowd at Martha's Vineyard when the unhappy First Couple arrived on holiday last August, gave the tabloids the opening they wanted. Not only was Chelsea now legally an adult, but she was taking a political role. She was fair game.
A barrier was broken by the tabloid New York Post, which printed a story in the autumn about Chelsea breaking up with her "first serious boyfriend" and seeking counselling as a consequence of her father's troubles. The broadsheet papers and television appeared embarrassed rather than liberated by People's decision to publish, burying reports of the Clintons' protest downpage.
The effect of the Lewinsky scandal on Chelsea has been hinted at by the President's half-brother, Roger, who said she had been "deeply shaken" by the scandal, "like Hillary".
The evening before her father's televised admission, she reportedly asked to see Rev Jesse Jackson, a regular visitor and counsellor to the Clintons, who spent the whole evening consoling her.