Johnson told Sky News: "For those of us who campaigned for Brexit and who believe in the opportunities of Brexit, it is bitterly disappointing."
Corbyn and his Labour Party have pushed to remain closely aligned with European rules and regulations after Britain leaves the continental trading bloc.
The Opposition Leader wants Britain to join a customs union with the EU, which would probably mean that it could not seek its own independent trade deals.
Seeking trade deals with the US and other global powers - and stopping free-flowing European migration to Britain - were May's top aspirations for Brexit.
The Prime Minister's gambit, to reach across to Corbyn, shows how dire May's position is.
She is a weakened leader, chairing a fractious Cabinet in open revolt, filled with feuding "remainers" and "leavers" who aspire to replace her. And she presides over a party that remains as divided as ever on Brexit, almost three years after the June 2016 referendum on Britain's membership in the EU.
Depending on one's vantage point, either the Conservative Party has failed May, or she has failed her party.
The Prime Minister negotiated, mostly in secret, for two years with her EU counterparts, to produce a withdrawal agreement that has been defeated three times in the House of Commons - and voted down by large numbers of Conservative MPs.
These very same Tories tried to oust their leader in a no-confidence vote. They finally extracted a promise from May that if she ever gets a withdrawal approved by the House of Commons, she will resign before the next stage of Brexit trade talks gets underway.
Brexit Secretary Stephen Barclay blamed hardline Brexiteers for the impasse. Barclay said Conservatives who refused to back May's deal in the House of Commons essentially forced her to negotiate with Corbyn.
Barclay told BBC Radio that the only alternative left was to look for support from Labour "because 35 of my own colleagues would not support the Prime Minister's deal" in the last vote.
Barclay said the result will probably be a "soft Brexit or no Brexit at all".
"It is undesirable, but it's the remorseless logic of the numbers of the House of Commons," he said.
Regardless, May's pivot threatens to implode the Conservative Party.
Conservative MP Nigel Adams quit the Government, saying he didn't agree that a deal "cooked up with a Marxist" like Corbyn was better than no deal.
A few hours later, Chris Heaton-Harris, a junior Brexit minister, resigned.
Heaton-Harris was the 30th minister to resign since May became Prime Minister in 2016. According to the Institute for Government, an independent think-tank, May has been handed more resignations than any other prime minister in the past four decades. During Tony Blair's 10-year-premiership, 29 ministers resigned.
In his letter to the Prime Minister, Heaton-Harris said he wanted Britain to leave the bloc on March 29, as promised, and could not support any further extensions. "Every time we seek an extension to this process we diminish faith in our political system," he wrote.
Others tweeted or gave interviews voicing frustration.
Iain Duncan Smith, a prominent Brexiteer and former leader of the Conservative Party, told the BBC that he was "absolutely appalled" at the idea of getting the deal through with the help of Labour votes.
Jacob Rees-Mogg, a leading Brexiteer, tweeted: "Remainers take charge to thwart the voters."
Fighting over Europe is nothing new for the Conservatives.
Such party divisions led to the downfall of several prime ministers, including David Cameron, John Major and Margaret Thatcher.
Cameron, May's predecessor, called the 2016 referendum to appease the Eurosceptics in his party and to outmaneuver the UK Independence Party, which wanted Britain to exit the bloc. Cameron thought he could win and his party would finally stop "banging on about Europe," as he memorably once said.
But since then, the banging has only become louder.
Anand Menon, professor of European politics at King's College London, said that "an awful lot" of the Brexit drama "has been about keeping the Conservative Party together."
He said there are intraparty politics in both main parties, and "you can't deny just how important that has been."
European diplomats in Brussels described a stomach-churning process that began with May's surprise that she would reach out to Corbyn and ended with some feeling that Britain's EU membership card was about to receive a long extension.
The first reaction was anger and frustration among Europeans at a perceived spurning of all of their red lines and a request to extend Britain's departure date to May 22 but no longer, diplomats said.
That deadline would create the possibility of a disorderly British crash-out the day before the start of crucial European Parliament elections, a possibility that EU leaders rule out because it could sway the polls.
But after putting May's speech on rewind and watching a few times attitudes started to soften. The reversal was complete after British officials passed the message to their colleagues today that May was prepared to hold European elections if she did not win approval of her divorce deal by next Thursday, when EU leaders are set to meet.
That would pave the way to a much longer extension, with terms defined by the remaining 27 EU leaders. Although that move is by no means a given, the British appear to be betting that the EU would rather have them inside as reluctant members than outside, with no transition period or safety net, which would spark economic chaos.
If, during the longer extension period, the British come up with a Brexit plan acceptable to both sides, so much the better, the Europeans say, and the departure date could be speeded up.
