She will also go over their heads by appealing directly to voters, saying: "In Parliament and beyond it, I will make the case for this deal with all my heart, and I look forward to that campaign."
Jeremy Hunt, the Foreign Secretary, issued a stark warning to rebel Tory MPs that if they rejected the deal the Government could collapse.
Arlene Foster, the DUP leader, admitted that her party had held talks with Cabinet ministers behind May's back about negotiating a Norway-style trade deal with the EU as a "plan B" if the current deal was voted down.
May will travel to every part of the UK to spread her message, starting tomorrow, while in the capital MPs will debate the deal for more than a week before the crucial vote.
On each stop of her journey, May will set out what she says are the key advantages of the deal.
Her themes include the ability of the UK to set its own migration policy, the "Brexit boost" to the economy after Britain stops paying billions a year to the EU and the protection of jobs and supply chains May says the agreement guarantees.
Insiders say that in the final days before the "meaningful vote" in Parliament next month, May favours holding a prime-time Sunday night televised debate with Corbyn, who called the deal a "miserable failure of negotiation".
If the Labour leader declines to agree to a debate, May will hold a Question Time-style session with a television audience hosted by David Dimbleby.
Downing Street plans for her to make a speech on December 11, the day before the vote, with the vote itself taking place the day before the next EU leaders' summit.
The timing of the vote will enable May to return to Brussels the day after her parliamentary vote and call on EU leaders to reiterate a message set out yesterday: that if the deal is rejected, a no-deal Brexit is the only alternative.
May said: "This is the deal that's on the table, it's the best possible deal, it's the only possible deal."
But Iain Duncan Smith, the former Tory leader, insisted the deal must be changed, saying all references to the so-called Northern Irish "backstop" must be removed, and Foster, on whose party's votes May depends for her majority, said: "I believe we must use the time now to look for a third way, a better way."
The Prime Minister is well aware that some MPs will hope that by voting against her deal, they can send her back to Brussels the next day to seek better terms.
However, Jean-Claude Juncker, the President of the European Commission, said that MPs who thought they could get a better deal by rejecting the one on offer would be "disappointed".
He told the BBC: "I am never changing my mind...if the House [of Commons] would say no, we will have no deal."
Speaking in the Commons tomorrow, May will say it is MPs' "duty" to "listen to our constituents" and deliver the Brexit the country voted for.
She will say that rejecting the deal will take Britain "back to square one", adding: "Before Christmas, MPs will vote on this deal. It will be one of the most significant votes that Parliament has held for many years. On it will depend whether we move forward together into a brighter future or open the door to yet more division and uncertainty."
May's campaign will start with a meeting of her Cabinet when she will urge them to help her win the argument.
She will then head to the House of Commons to make a statement about the EU's agreement to the deal before she makes the regional visits.
One senior aide admitted it will appear like a "mini-election campaign" adding: "The Cabinet and the Prime Minister will be taking the message around all four corners of the United Kingdom over the period of the campaign."
The Government is expected to publish Treasury forecasts on Thursday about the economic impact of a deal against leaving the EU without a deal.
On Friday, May is expected to be questioned by senior Tory MPs on the House of Commons liaison committee before heading to Argentina for three days for the G20 meeting of world leaders.
Aides to both May and Corbyn left open the door to debating the deal live on television.
A Number 10 official said "nothing has been decided" on whether to take part in a debate. An aide to Corbyn said his office would consider the offer of a debate if it were made.
The Labour leader said the party would oppose the package in Parliament and work with MPs from other parties to block a no-deal Brexit.
Corbyn said: "This is a bad deal for the country. It is the result of a miserable failure of negotiation that leaves us with the worst of all worlds.
"It gives us less say over our future, and puts jobs and living standards at risk."
