Transcripts of candid conversations between the former US president and the ex-UK prime minister have been published following a BBC FOI request.
They start with Mr Clinton congratulating Mr Blair on his 1997 general election victory.
Soon afterwards, Mr Blair is already targeting a second term in office.
"We are having a hell of a time getting under way - the honeymoon doesn't last long," he said as Mr Clinton attended a meeting of his new Cabinet.
"The second term is what we need; to take nothing for granted and not rely on the other side's unpopularity."
In a conversation straight after Labour's election victory, Mr Clinton commiserates with Mr Blair for "getting so much grief with the press".
"Always the Right attack you and the Left don't defend you," Mr Blair replies.
The transcripts of calls and meetings between the two leaders also cover the Northern Ireland peace process and the death of Princess Diana, as documented in a blog by BBC Freedom of Information specialist Martin Rosenbaum.
They have been released by the Clinton Presidential Library, which stores the records of his presidency.
In one conversation, Mr Clinton praises Gordon Brown, then Mr Blair's chancellor, at a meeting of international finance ministers.
"He was terrific, you would have been really proud of him," he says.
"Bloody bright, isn't he?" Mr Blair replies.
When the US president speaks of his affection for Durham Cathedral, Mr Blair says: "I was a choirboy at the cathedral in Durham, if you can believe that."
"You still have that choirboy look," replies Mr Clinton.
Towards the end of the exchanges, in 2000, Mr Blair is preparing for the birth of his fourth child. and Mr Clinton is soon to leave office.
"You know, after January I'm available for babysitting duties," he says.
"I wouldn't say that or you'll be doing it," Mr Blair replies.
"You said you wanted to continue my work with the Third Way," Mr Clinton says, "and this is it:  helping Blair balance life and family".
Bill Clinton told Tony Blair he had "that choirboy look" - and offered to babysit for his children when he left office, newly released records show.