Documents released by Clinton's presidential library show the two leaders conferring on world events and comparing notes on how to balance being a world leader with bringing up a family.
The 500 pages of the documents begin with Clinton calling the new prime minister hours after his 1997 victory and end in late 2000 as the president was preparing to hand over the White House to George W Bush, the Daily Telegraph reported.
Both men express concern for Diana's two young sons, William and Harry.
Blair says: "She was such a rock of stability in the sense she connected them with the outside world. The eldest boy, William, is quite like her in a way, he is very 'feet on the ground'. He does things kids his age do."
"I just feel so bad for her. She was just basically getting a hold of her life," Clinton says.
As Blair's wife Cherie prepares to give birth to his son Leo, Clinton greets him by saying: "Hello, dad!"
Blair goes on to describe how Cherie is "in great form but just keeps getting bigger and bigger," and how he felt as though "my life's about to begin again."
Clinton, whose time in office is drawing to a close, offers the Blairs a helping hand.
"You know, after January I'm available for babysitting duties," he jokes.
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