The newspaper said campaigners will write to the UK government on December 13 saying they are taking it to the High Court in an effort to keep Britain in the single market.
It said the claimants will seek a judicial review in an attempt to give lawmakers a new power of veto over the terms on which Britain leaves the EU.
They argue the government “has no mandate” to withdraw from the single market because it was not on the referendum ballot paper on June 23 and was not part of the ruling Conservative Party's manifesto for the 2015 general election.
May has said she wants to invoke Article 50 of the EU's Lisbon Treaty by the end of March, kicking off up to two years of exit negotiations.
However the High Court ruled last month that Article 50 cannot be triggered without parliament's assent. That ruling is being challenged by the government in Britain's Supreme Court.
The Sunday Times said the new court case hinges on whether the government would also have to trigger another legal measure — Article 127 of the European Economic Area agreement — in order to quit the single market.
It said ministers argue Britain automatically exits the single market when it quits the EU. But, it said if the claimants win the new case, the government would have to gain the approval of lawmakers.
The Department for Exiting the European Union could not be immediately reached for comment.
A group of British and Irish lawyers are also seeking to challenge Britain's decision to leave the EU in the Irish High Court to try to establish if Brexit can be reversed once divorce talks have been triggered.
Pro-Brexit critics have cast the legal battles as an attempt by a pro-EU establishment to thwart the result of the referendum, when Britons voted by 52-48 percent to leave the EU.