British Prime Theresa May has narrowly avoided a humiliating parliamentary defeat on Brexit after making a major concession to Conservative MPs that could hand them an effective veto on her deal to leave the European Union.

In a day of drama at Westminster, Ms May caved into the rebel demands for a “meaningful vote” on Brexit in order to prevent a defeat on the EU Withdrawal bill which could have triggered a leadership crisis for Ms May.

The vote on Wednesday morning (AEST) means Ms May’s plans to end more than 40 years of membership of the EU were still on track, but her concession to Parliament means that MPs now have more power if she fails to secure a Brexit deal, which may lead to a softer approach to Britain’s European divorce.

Leading Conservative rebels welcomed the “important concessions” from the government, but insisted that ministers must follow through on their concession or face a defeat when the bill returns to the House of Commons later this month.

Ms May’s leadership has come under increasing pressure as she struggles to move ahead with all-but stalled Brexit talks in Brussels.

The new concession is likely to anger Brexit-supporting MPs, some of whom have told UK media they intend to push for Ms May’s removal if she “backslides” any more on Brexit.

Earlier, Brexit minister David Davis told parliament a government defeat would undermine negotiations with Brussels and warned lawmakers the government would never allow them to “reverse Brexit”.

“The government cannot demonstrate the flexibility necessary for a successful deal if its hands are tied midway through that process,” he said.

Ms May has come under increasing pressure over stalled Brexit talks. Photo: Getty

The victory was the first major win in two days of debates on the government’s EU withdrawal bill, which will sever ties with the EU, after the upper house of parliament, the House of Lords, introduced 15 changes.

In the tense atmosphere where it was not clear which way the vote would go, the government secured its victory only after offering concessions to one of the leaders of a group of Conservative lawmakers who were threatening to vote against Ms May.

Earlier, Ms May appeared to have also stemmed a rebellion over her commitment to leaving the EU’s customs union which will transform Britain’s trading relationships for decades to come.

Lawmakers will vote on Wednesday local time on whether the government’s move to dismiss a House of Lords amendment requiring ministers to report what efforts they had made in negotiations to secure a customs union by the end of October.

Instead, the government has proposed reporting its efforts to secure a customs arrangement.

In a day of drama, Ms May’s position seemed suddenly weaker when junior justice minister Phillip Lee, who has long been critical of the government’s Brexit strategy, resigned and said he would vote against the government.

But it has secured victory on all the other votes so far on Tuesday, including a challenge to the government’s plan to put March 29, 2019, or ‘Brexit Day’.

-with AAP