The Conservatives had dropped almost 1,300 seats and lost control of 45 councils after the final results came in - the worst performance, in raw numbers, by a governing party in local elections since 1995.
However, many Conservative eurosceptics fear that the newly launched Brexit Party of veteran anti-EU campaigner Nigel Farage, which did not contest the local polls, will cost them support in the European elections later this month.
Theresa May must set a date for her departure or her MPs will do it for her, a former Conservative Party leader has said in the wake of "devastating" local election results.
She also stressed the Brexit chaos had caused very unusual times politically, telling Sky News: We have to be honest, parliament is a mess at the moment.
Labour has demanded guarantees on workers' rights and a permanent customs union with the EU as a condition for supporting an EU withdrawal deal.
"If we can find a solution that delivers the benefits of the customs union without signing up to the current arrangements, then I think there will be potential (for a deal)".
Hancock suggested there could be greater willingness to compromise following the local election losses.
"The glimmer of hope we have in this situation is that both Conservative core voters and Labour core voters want Brexit sorted, and both would be extremely angry with the party they voted for if we had another general election without Brexit being delivered", he told the Press Association news agency.
Pelosi Weighs Holding Barr In Contempt for Refusing House Dems' Demands
Attorney General William Barr started off the day by not appearing for his previously scheduled testimony with House Democrats. McGahn also said that the president subsequently wanted him to make a statement that the report about the incident was untrue.
The Prime Minister appeared to put the future of her party in the hands of Labour leader Jeremy Corbyn, by welcoming his offer to help her resolve the Brexit deadlock.
But Labour had not benefitted from the Tory losses, the big gains going to Lib Dems, Greens and independents.
Asked who was responsible for the losses, he told reporters in Africa: "You can look at lots of different groups of people - you can look at Brexit purists in my party who have consistently refused to compromise and put Brexit in peril".
"In the last week government ministers and officials presented Labour with a new offer on a customs arrangement that would effectively see the United Kingdom remain in the key aspects of a customs union with the EU", the sources familiar with the talks said.
The website said one source had told it that "the offer would be tantamount to the government accepting in full Labour's demands".
However, the sources did not think a deal was necessarily imminent, as Labour might wish to delay any agreement until after the European elections to maximise the damage to May.
It is unclear if the European Union would approve a temporary customs deal, as border controls might later be needed between Ireland and Northern Ireland if the deal broke down.
And even if May and Corbyn agree, there is no certainty they could convince enough lawmakers in their parties to ensure a majority for the deal.