The Prime Minister is facing a series of legal challenges over his decision to suspend Parliament for up to five weeks ahead of a Queen's Speech on October 14.
Former Conservative leader Sir John will now seek to join an action being brought by campaigner Gina Miller.
Sir John suggested his experience in Number 10 could assist the High Court in deciding whether Mr Johnson's actions in proroguing Parliament are lawful.
Deputy Labour leader Tom Watson said he was also joining the legal action against what he called "an unprecedented affront to democracy".
But Mr Johnson defended his decision and warned efforts to frustrate Brexit on October 31 would be seized on by Brussels to avoid offering a good deal.
"I'm afraid that the more our friends and partners think, at the back of their mind, that Brexit could be stopped, that the UK could be kept in by Parliament, the less likely they are to give us the deal that we need," he told Sky News.
He also said there would be a backlash if people's votes in the 2016 referendum were not respected.
"If we frustrate that mandate, if we stop the UK from leaving on October 31, if that's what parliamentarians end up doing, it will do lasting damage to people's trust in politics.
"It will do lasting and catastrophic damage to the major parties in this country and I think this political generation won't be forgiven for failing to honour that promise."
Businesswoman Ms Miller - who previously took the Government to court over the triggering of Article 50 to start the Brexit process - said her case would be heard on September 5.
Opponents claimed prorogation was aimed at stopping discussion of Brexit and hampering cross-party efforts to block the prospect of a no-deal withdrawal from the European Union - an allegation denied by Mr Johnson.
"In view of the imminence of the prorogation - and to avoid duplication of effort and taking up the court's time through repetition - I intend to seek the court's permission to intervene in the claim already initiated by Gina Miller, rather than to commence separate proceedings," Sir John said.
"If granted permission to intervene, I intend to seek to assist the court from the perspective of having served in Government as a minister and prime minister, and also in Parliament for many years as a member of the House of Commons."
In a separate legal case in Scotland, judge Lord Doherty rejected a call for an interim interdict to block the suspension of Parliament but said a full hearing would take place on Tuesday.
Meanwhile, opponents of Mr Johnson's strategy appear increasingly confident of finding a way to block a no-deal Brexit despite the prospect of Parliament shutting down temporarily.
Tory rebel ringleader Sir Oliver Letwin said he had been in talks with Speaker John Bercow about the parliamentary procedures that will apply.
The former minister told BBC Radio 4's Today: "I know that there are a number of my colleagues who feel as I do, that a disorderly no-deal exit is a very bad idea, and they have in the past been willing to come and support efforts to prevent that happening and I very much hope that will happen again."
Sir Oliver said the move could force Mr Johnson to delay Brexit beyond the October 31 deadline unless there is a Withdrawal Agreement with Brussels.
On the Labour side, shadow attorney general Baroness Chakrabarti said "since the constitutional outrage" she had "greater comfort that minds are now focused, especially on the Conservative side".
She told Today there were ways of preventing filibusters and "any sort of public school dirty tricks" aimed at blocking legislation when it reaches the Lords.
Foreign Secretary Dominic Raab dismissed as "nonsense" the suggestion that prorogation was a constitutional outrage.
The prospect of an explosive Commons battle next week came as Mr Johnson called for both the UK and EU to "step up the tempo" in talks with negotiators sitting down twice a week in September.
But Ireland's deputy prime minister, Simon Coveney, said so far the UK had not put forward any "credible" alternatives to the backstop - the contingency plan aimed at preventing a hard border with the UK.
Brussels again demanded "concrete proposals" that were "compatible with the Withdrawal Agreement".
Labour leader Jeremy Corbyn encouraged his MPs to join public protests against a no-deal Brexit.
He said: "There are also public protests across the country this Saturday, there will be a rally in Parliament Square on Tuesday evening, and I encourage Labour MPs to be present and to share our message."