The letter argues that both the government's current plans for Brexit, and a no-deal Brexit, would leave the country worse off than they were being in the European Union if the country left in March.
However, in order for the EU to accept this compromise, Mrs May must convince her colleagues and the DUP to agree that Northern Ireland can only opt out of the customs union when another mechanism of ensuring a soft Border is agreed.
Mr Lidington replied by saying: "We're certainly. very close to resolving it".
Her plan would avoid the need for a hard border in Northern Ireland, which has always been the main sticking point in negotiations.
The Sunday Times report said preparations for a final Brexit deal were "far more advanced than previously disclosed" and that May's agreement would satisfy both remain-voting Tories and the hardline Eurosceptics within her party.
The newspaper also said May is on course to secure an agreement on a "future economic partnership" that would let Britain keep open the prospect of a free trade accord similar to the one Canada has with the EU.
It is speculated that May is hoping for enough progress in Brexit talks this week to secure a summit later this month in which the final details of a deal will be negotiated.
However, if these Brexiteers are not convinced that the customs union is only a temporary measure they would nearly certainly rebel.
The EU has proposed a backstop that would mean Northern Ireland staying in the EU customs union, large parts of the single market and the EU VAT system.
"We are being asked to choose between the break-up of the Union - at least for economic purposes - or the subjugation of the whole country", said the Uxbridge MP. "We need to stop before it is too late".
"The prime minister has been clear that we are making good progress on the future relationship, and 95 percent of the withdrawal agreement is now settled and negotiations are ongoing".
The Taoiseach also said the UK's withdrawal from the bloc had undermined the historic Good Friday Agreement that ended three decades of violence in Northern Ireland and called for a guarantee that there would be no return of a hard border with the Republic.
According to the Telegraph, May has included Attorney General Geoffrey Cox, the government's chief legal adviser, in her special Brexit Cabinet after ministers said they wouldn't sign off on a deal without his advice.
A backstop arrangement is thought to be the major obstacle to the United Kingdom and European Union agreeing a Brexit deal that would allow negotiations to begin on a future trading relationship.
"We hope a deal can be done but we're not there yet".