Mrs May has reportedly secured concessions from the EU to avoid a hard border in Ireland, the Sunday Times reports.
The agreement would avoid the need for an Irish backstop – which has left Britain and Brussels deadlocked – and would stop Northern Ireland being treated differently to the rest of the UK.
But Downing Street has reportedly dismissed the report as “speculation”.
According to the Sunday Times, the agreement would include an “exit clause” aimed at convincing Brexiteers that remaining in the customs union would be temporary.
“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”
Downing Street spokesperson
Preparations for a final deal were far more advanced than previously disclosed, the report said.
Mrs Mays cabinet are set to meet on Tuesday to discuss her plan, in the hope that enough progress would be made by Friday for bloc chiefs to announce a special summit, it has been claimed.
A spokesperson for the PM said: “This is all speculation.
“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.”
Mrs May is said to continue to be pushing for a Canada-style free trade deal with the EU.
But she is expected to warn Brexiteers that if they reject the deal she puts on the table, they will be faced with a very difficult no deal scenario.
It comes as dozens of business leaders have demanded a public vote on the final terms of our exit from the European Union, amid fears the country faces either a “blindfold or destructive hard Brexit”.
Former Sainsburys boss Justin King and ex BT group chairman Sir Mike Rake are among the business heavyweights who have signed a letter demanding the people have their say.
The UK is due to leave the EU on Brexit D-Day, March 29, 2019.