The party’s Brexit shadow secretary, Sir Keir, urged Tories to support the change to the legislation, which will aim to protect devolved powers after Britain leaves the EU.
The EU Withdrawal Bill is due to begin its final stage in the House of Commons on Tuesday without changes to a clause governing devolution.
The Brexit Bill is crucial to enshrine all EU laws into domestic legislation to facilitate the country’s post-Brexit legal transition.
It folows Scottish First Minister Nicola Sturgeon yesterday revealing the SNP will oppose the bill in its current form, calling out Clause 11 of the legislation as a power grab.
The contentious clause means all devolved powers returning to the UK from Brussels will initially go to Westminster.
Labour’s Shadow Scottish Secretary Lesley Laird and Shadow Brexit Secretary Sir Keir have written to Scotland’s Tory MPs urging them to vote for Labour’s amendment.
Ms Laird said: “The EU Withdrawal Bill as it stands drives a coach and horses through the devolution settlement. That is simply unacceptable.
“The Secretary of State has broken his promise that this legislation would be fixed to protect the principles of devolution.
“The Bill can still be changed before it goes to the House of Lords – Labour’s amendment today can fix these problems.
“That’s why Labour has written to Scottish Tory MPs urging them to back our amendment – and protect the devolution settlement.”
Mrs May’s government claims the move is needed to set up UK-wide frameworks in some areas and it will help devolved further power to Holyrood.
But Scottish and Welsh administrations are refusing to grant legislative consent for the Bill until the clause is amended.
Scottish Conservative constitution spokesman Adam Tomkins has said the Bill needs to be amended to comply with the devolution settlement.
A UK Government spokeswoman said: “We want all parts of the UK to come together in support of this legislation, which is crucial to delivering the outcome of the referendum.”