Mr Johnson is facing mounting criticism over the Internal Market Bill, as leaders from across the political spectrum attack the threat to break international law.
Senior Conservatives are not backing down on their rebellion against the Prime Minister’s plans despite his warning that Brussels could “carve up our country” without the Bill.
Sir John and Mr Blair said the legislation imperils the Irish peace process, trade negotiations and the UK’s integrity.
Sir Keir Starmer accused Mr Johnson of “reigniting old rows” by working to flout his own withdrawal agreement, but pledged Labour support if he addresses “substantial” concerns.
Tory rebels suggested their numbers were growing and opinions were only hardened by Mr Johnson’s increased rhetoric that the EU could impose a trade border in the Irish Sea.
Sir John and Mr Blair, the former Conservative and Labour leaders, united to condemn Mr Johnson’s controversial UK Internal Market Bill in an article for the Sunday Times.
“We both opposed Brexit. We both accept it is now happening. But this way of negotiating, with reason cast aside in pursuit of ideology and cavalier bombast posing as serious diplomacy, is irresponsible, wrong in principle and dangerous in practice,” they said.
“It raises questions that go far beyond the impact on Ireland, the peace process and negotiations for a trade deal — crucial though they are. It questions the very integrity of our nation.”
Sir Keir called on Mr Johnson to throw out clauses that could breach international law and those that lead the devolved administrations to warn of a “power grab” to get Labour support.
“If the Government fixes the substantial cross-party concerns that have been raised about the Internal Market Bill, then we are prepared to back it,” he wrote in the Sunday Telegraph.
“But if they do not, and the talk collapse, then it is their failure and incompetence that will have let the British people down.”
The newspaper also reported that ministers are considering opt-outs from the Human Rights Act, and that a formal review could be announced in the coming weeks.
The Tory manifesto pledged to “update” the Act after Brexit, but the move is contentious with Brussels and negotiators have expressed concern over the UK’s commitment to the European Convention on Human Rights.
Justice Secretary Robert Buckland, who is under pressure to explain how Government plans to override elements of the Brexit deal are consistent with obligations under international law, will face questioning from the media on Sunday.
His Labour shadow, Lord Falconer, said: “A future where (the) UK breaks its international law obligations, and opts out of Human Rights protections is a very bad future.”
Despite Mr Johnson’s attempts to drum up support, Tory chairman of the Commons defence committee Tobias Ellwood said on Saturday that he could not support the legislation without it being changed.
“Already this Bill is damaging brand UK, diminishing our role-model status as defender of global standards. As we go to the wire, let’s see more British statecraft – less Nixonian Madman Theory,” he tweeted.
Commons justice committee chairman Sir Bob Neill, who has tabled an amendment which he said would impose a “parliamentary lock” on any changes to the Withdrawal Agreement, said he still contends it contains “objectionable” elements.
Damian Green, who was Theresa May’s deputy when she was prime minister and is backing the amendment, was also understood to not have been won over by Mr Johnson’s argument.
Sir Roger Gale also remained a vehement critic, telling Times Radio: “If anybody is responsible, if it happens, for bringing the union down, it will be (chief aide Dominic) Cummings and Mr Johnson.”