The talkRADIO host told listeners: "Working class Labour voters in key seats voted to leave", she said.
Corbyn, a lifelong eurosceptic who supports Brexit, has been highly ambiguous throughout the negotiations even though many of his MPs support a second referendum.
Corbyn has been under pressure for some time to support a second referendum.
"The prime minister's botched deal provides no certainty or guarantees for the future", he said, accusing May of running down the clock in a "grotesquely reckless" way.
Labour is not yet making clear what its proposed referendum would be on.
This debacle has finally revealed where the power lies in the Cabinet: with a set of Tory MPs who in all respects would much prefer Britain stay in the EU.
The PM's announcement came amid mounting expectations of a ministerial revolt, with three members of the Government - Richard Harrington, Claire Perry and Margot James - breaking ranks to warn in a newspaper article that they would back a cross-party bill created to prevent a no-deal Brexit.
With just over a month until Britain is due to leave the bloc, Prime Minister Theresa May is seeking changes to her exit deal in order to break an impasse in parliament.
Businesses and the markets breathed a sigh of relief at May's statement, which did not rule out "no deal" but at least pushed it a bit further away.
Mr Tusk said delaying the UK's withdrawal beyond March 29 is now a "rational solution", warning that the only alternative, if MPs can not agree a deal, is "a chaotic Brexit". Such an arrangement would likely mean that Britain would have to continue to accept the free flow of immigrants from Europe.
Speaking to the BBC, and reiterated on Sky News, he said: "The basic choice needs to be between a credible Leave option and Remain".
May returned on Monday night from an European Union summit in Egypt where one European Union leader after another asked her whether she would delay Brexit.
However, it will give her ammunition to persuade hardliners who consider her deal a travesty that they could lose Brexit altogether if they don't back it when it returns to Parliament.
He called on May to rule out a "reckless, cliff-edge" Brexit.
The former Secretary of State for Work and Pensions also said: "It usually works like this: you would speak and see the Prime Minister in private to tell her what you are going to do".
Mrs May's Brexit deal was comprehensively rejected by MPs on 15 January and she has said they'll get a second chance to vote on it - possibly with some changes - by 12 March.
May has said a new vote on any revised Brexit deal won't be held this week and could come as late as March 12.
May then rejected proposals of a second referendum, saying it would take the country "back to square one".
So she vowed to press on with the search in Brussels for a compromise to her own Brexit deal.
Will there be an option to remain in the EU?
MPs and peers are told to report any complaints to Ms Formby but Mr Watson said "a number of colleagues" had told him their concerns were not being dealt with in an "adequate and timely manner". I've had a real sense from the meetings I've had here and the conversations I've had in recent days that we can achieve that deal.