The Anglican leaders urged politicians and the public to insert civility into the debate amid intense discussions over Prime Minister Theresa May's Brexit deal.
Blair, the former leader of the Labour Party, now in opposition to the Conservative Party, made his comments in London at an event organized by the "People's Vote" campaign group, which was pushing for another vote on the UK's withdrawal from the EU, Efe news reported.
Mr Clark also cautioned against a second referendum, saying it would "continue the uncertainty for many more months".
"It's clear the Prime Minister has failed to renegotiate her deal, failed to get any meaningful reassurances - there is no excuse for any more dither or delay".
But she added: "The EU is clear, as I am, that if we are going to leave with a deal, this is it".
In a pointed swipe at the Labour heavyweight, the Prime Minister said a second referendum would amount to Parliament abdicating responsibility.
May's chief of staff, Gavin Barwell, denied reports from two Sunday papers that he was among those who believed a second referendum was the only way forward in a series of tweets.
We had the people's vote, we had the referendum, and now we've got to get on with implementing it.
Dame Margaret said: "It is highly significant that Downing Street felt it had to issue these advance extracts of her statement to the House of Commons on Sunday night, because officials know the prospect of a People's Vote is being discussed not just in Westminster but in the corridors of Whitehall too".More news: Premier League best bets: Everton can cause problems for Manchester City defence
May's office said she will tell lawmakers in the House of Commons Monday afternoon that staging another referendum "would say to millions who trusted in democracy that our democracy does not deliver".
Britain's Foreign Secretary Jeremy Hunt said parliament could still rally behind May's agreement with additional assurances and said such clarifications were likely because European Union countries knew no deal would be a disaster for them.
With just over 100 days until Britain leaves the bloc on March 29, Brexit remains up in the air with growing calls for a no-deal exit, a potentially disorderly divorce that business fears would be highly damaging, or for a second referendum.
Speculation has intensified about a second referendum on leaving the European Union since May withdrew a House of Commons vote on her deal with the bloc last week after it became clear it was headed for defeat.
The Liberal Democrats - a minor opposition party previously in a coalition government with May's predecessor David Cameron - have come out in favor of a new vote, but this has not translated into much of an increase in support.
The EU27 have said they will do their utmost so that the backstop - meant to ensure there is no return to a hard border between Northern Ireland and the Republic - will not be needed.
Scottish First Minister Nicola Sturgeon, leader of the Scottish National Party, accused Labour, which is divided over what tactics to pursue, of holding up the drive for a second referendum by failing to table a no-confidence motion.
"And let us not follow the leader of the opposition in thinking what gives him the best chance in forcing a general election, for at this critical moment in our history we should be thinking not about our parties' interest but the national interest", May said. He said the country had been "held hostage" by division in the Tories, but said the issue was such that it would not make any difference if it was a Labour or Conservative government "or a divine government" running the negotiations.