A report says an all-UK customs deal will be written into the legally binding agreement governing the UK's withdrawal from the EU.
The agreement will include an "exit clause" designed to convince Brexiteers that remaining in the customs union is only temporary, the newspaper said.
Preparations for a final deal are also reportedly "far more advanced than previously disclosed", and will lead to a document of around 50 pages being published.
The cabinet will meet on Tuesday to discuss Mrs May's plan, and she hopes there will be enough progress by Friday for the European Union to announce a special summit, the report said.
He said: "The prime minister has been clear that we are making good progress on the future relationship and 95% of the withdrawal agreement is now settled and negotiations are ongoing."