"Any debate must involve someone who believes in Brexit & the British people being fully in control of their laws, rather than giving back control to the European Union like the PM's deal".
His intervention came after Mrs May had rejected calls for leaders of smaller political parties to join the two main party leaders in a televised debate.
A number of dates were being discussed, including December 9, which could leave the leaders in a ratings battle with the final of reality television programme I'm A Celebrity.Get Me Out Of Here!.
She added: "Exactly how it might be done, if he takes it up, would be a matter for the broadcasters to determine".
Such is the pressure on her leadership that The Times newspaper reported some Brexit-supporting MPs in May's Conservative Party were demanding that she set out when she will quit as a condition for supporting the deal.
"This is a really important moment for our country".
If it goes ahead, it will be the first time the pair have gone toe-to-toe in a USA presidential-style debate as May refused to face off against Corbyn ahead of the 2017 general election. It is a big decision MPs will be taking.
Asked today whether she had considered holding a bushtucker trial instead, May replied: "I think this is an issue on which we want to debate the questions of our future".
The prime minister will attempt to rally support from MPs on Monday in a Commons statement, in which she will urge them to get behind her or risk "more division and more uncertainty", Politics Home reports. "I believe in a better way forward".
"I am willing to stand up and explain why I think it is the best possible deal available for the United Kingdom", she told The Sun.
"I think what is important when MPs come to vote is that they think about the national interest - that means delivering on Brexit but it means doing it in a way that delivers for people in protecting their security, protecting their jobs, protecting their livelihoods".
Liberal Democrat leader Vince Cable has demanded to be included in any debate as have leaders of Scottish and Welsh nationalist parties.
Challenged on BBC's Andrew Marr Show about the prospect of an election being unlikely, Mr Burgon said: "I think the age of the experts is over".
41% say they would like politicians to vote for the PM's deal and 38% want them to vote against.
The Labour boss has said he would "relish" the chance for debate - but would wait until a formal invite comes in first.