A man holding a British national flag demonstrates in London, March 30, 2019. (Photo: VCG)

Britain's governing Conservatives need to be open to compromise with the opposition Labour Party in order to deliver Brexit following heavy losses in Thursday's local elections, senior ministers said on Saturday.

Prime Minister Theresa May's Conservatives lost 1,332 seats on English local councils that were up for re-election and Labour, which would typically aim to gain hundreds of seats in a mid-term vote, instead lost 81.

Many voters expressed frustration at May's failure to have taken Britain out of the European Union, almost three years after the country decided to leave in a referendum.

Following the election losses, May said on Friday that the message for both the Conservatives and Labour from Thursday's elections was that voters wanted parliament to deliver Brexit.

In a rare agreement, Labour leader Jeremy Corbyn said there was now a “huge impetus” on every lawmaker to get a Brexit deal done.

Foreign Minister Jeremy Hunt also said he saw a "glimmer of hope" that a compromise was possible, but said Labour's customs plans could not be a long-term solution for Britain.

Health Minister Matt Hancock gave a similar message in a BBC radio interview, "I think we need to be in the mood for compromise," he said, suggesting that there could be greater willingness to compromise following the election losses.

May on Saturday said she was optimistic she could reach a deal with Labour soon.

Behind closed doors, the government had already compromised on a customs union, reported Buzzfeed News.

"In the last week government ministers and officials presented Labour with a new offer on a customs arrangement that would effectively see the UK remain in the key aspects of a customs union with the EU," sources familiar with the talks told BuzzFeed News.

The website said one source had told it "the offer would be tantamount to the government accepting in full Labour's demands."

"We are getting closer and closer. There is not that much between the two main parties as I understand it within the room," Ruth Davidson, the leader of Scotland's Conservatives, told reporters at a party conference in Aberdeen.

"There is a deal that could be done in the next few days in quite short order, and I really hope we can get to that point," she said.

May opened talks with Labour a month ago in an attempt to break the impasse over Brexit after her deal to leave the European Union was rejected three times in parliament, but the talks didn't go smoothly since the two parties hold different stances on the proposal of the customs union.