The deadline to conclude the Oslo deal was May 1999, by which time a comprehensive agreement was meant to have been reached.
The president's frustration and sense of desperation as the deadline neared was clear in October 1998, when Mr Blair said that he had seen "Mr Arafat on his way to you," adding: "He still thinks it is possible to do a deal."
Mr Clinton replied: "I think it is. He wants it, I think. What we have to do is keep them from falling into these old habits. I keep telling them if we don't make this deal now, we've got May 1999 staring us straight in the face and I think it will be a disaster.
"It will be impossible to avoid a bad outcome unless we get this interim agreement now and get into final status talks."
He added that the process had become more difficult because the Palestinians' economic situation was "worse than when they signed on to peace.
"The Israelis close borders at the drop of a hat and they never concluded all these economic things, the roads and airport and industrial park, they have been on the verge of doing for months and months.
"Somehow we have to convince them they have to resolve all that crap. There is a train wreck facing them in May 1999 unless we somehow can give them some breathing room, some progress and confidence in one another's work."
Mr Blair was also keen to secure a deal, telling his counterpart: "I will be here and anything I can do to be helpful I will, just give me a call any time. I will speak to the Europeans or anything."
In June 1999, as the war between Yugoslavia and Kosovo nationalists which killed thousands of civilians was drawing to a close, Mr Clinton talked about sending his advisor, Holocaust survivor Mr Wiesel, to the area.
"I sent him to some of the camps, and he said, 'boy, there is so much anger there. Every Kosovar believes he knows the Serb who ousted him'."
During one call about Mr Wiesel and Kosovo the line failed. When they were reconnected the men shared a joke, with Mr Blair laughing that he would have the person responsible "executed".
The president discussed Shoah survivors again in 2000, chastising the media for how they had been treated during the US election.
After Mr Blair told him that "the right-wing press here has just been desperate for Bush," President Clinton responded: "They are all upset that the military people got disenfranchised because they came out in force for Governor Bush so he carried them two to one.
"But World War Two veterans and Holocaust survivors, they were perfectly fine with having them disenfranchised."
He added that right-wing media outlets "hate us more than we hate them. It's all about power to them. I will tell you more when I see you. I have not said publicly what I just told you."
During the fallout from the closest election ever, the result of which came down to less than 600 votes in Florida, the state's Jewish votes for nationalist candidate Pat Buchanan came under scrutiny.
Mr Clinton told Mr Blair that there had been "3,200 Jewish people who voted for Buchanan and we had a statistician say it was a trillion to one shot that Buchanan could have gotten that many votes".
One of the president's comments also foreshadowed Iran's rebellion against laws over nuclear nations and the subsequent rise in its nuclear capabilities.
"They say in the Middle East we are hypocrites," Mr Clinton said.
"They say there are others who have this, they say the Israelis, but no one else has shown a willingness to use them or share them with anyone else."
After Ehud Barak became Israel's Prime Minister in 1999, Mr Clinton said he had enjoyed "a great talk" with the new leader who, he told Mr Blair, "admires you so much".
In their last recorded conversation on the peace process in November 2000, Mr Clinton said: "The Israelis have good substantive policies, but the level of misunderstanding between the two sides is pretty high."
