Major, who led Britain from 1990 to 1997, said he feared the country was on course for a disorderly Brexit, with May unable to reconcile the two wings of the Conservative Party.
There is also speculation that former foreign secretary Boris Johnson could deliver a resignation speech.
In the past 18 months, said Johnson, "it is as though a fog of self-doubt has descended".
"It is not too late to save Brexit", Johnson said in his first statement to parliament since his resignation. We have changed tack once and we can change again.
He said Britain would end up stuck in a "miserable limbo" under the Chequers plan for future links with the European Union that prompted him to quit.
If the new clause, put forward by Conservative MP Stephen Hammond, has succeeded, it would have meant the British government having to negotiate Britain's participation in a custom's union with the EU if it couldn't agree a free trade area deal with Brussels.
May denied a suggestion in parliament that her Brexit plan was dead, and her spokesman said the decision to accept those amendments were "consistent" with the white paper policy document ministers agreed earlier this month. "Worst of all we allowed the question of the Northern Ireland border to dominate the debate".
Hard Brexit-backing Tory William Rees-Mogg praised Mr Johnson's words as "the speech of a statesman".
Rose fires 64 at Open to get back in contention
He then missed a great chance to hit the front on his own at the 5th, pushing a straightforward six-foot putt to the right of pin. McIlroy was within two shots of the lead until bogeys on two of his last three holes for a 70 left him four behind.
Labour's Wes Streeting called the speech a "total damp squib".
The pressures building on the prime minister have come to a head in the last week, after the government finally published its strategy for leaving the European Union, which disappointed both sides of the divide.
The latest squabbling over Brexit has renewed speculation about a possible leadership challenge for May, who has had a particularly rough few weeks with high-profile resignations, knife-edge votes in the House of Commons, and criticism of her Brexit plans from both those who want to remain in the European Union and those who want to leave.
"Failure to keep our promise to the electorate will nearly certainly lead to the catastrophe of Jeremy Corbyn (Labour leader) becoming Prime Minister and I can not sit back and allow that to happen", he said.
"At absolutely no point because BREXIT continues to mean BREXIT", Mrs.
May won the knife-edge vote by 307 votes to 301, highlighting the deep division in parliament and within her own party, and raising questions about whether May can have enough MPs to ratify the final Brexit deal.
Ms Soubry told BBC Radio 4: 'Your listeners should be angry that the likes of Jacob Rees-Mogg are running our country at this the most hard of times since the Second World War'.
Tuesday was also the second day running May faced revolt from backbench Tory lawmakers, after they failed on Monday by three votes to enact changes to a crucial post-Brexit customs bill.