The government is facing pressure to move toward Labour's position on a customs union, following a warning from a senior shadow minister - but the issue of Brexit was largely sidestepped in Prime Minister's Questions.
Instead of discussing Brexit in front of an usually empty House of Commons, the two party leaders sparred over social mobility, apparently saving the topic for the private negotiating table.
Shadow business secretary Rebecca Long-Bailey - someone who sits at that table - said the government may have no option but to give ground on key Labour demands if cross-party talks aimed at ending the deadlock were to succeed.
Her comments came amid reports that Environment Secretary Michael Gove had warned ministers they may have to offer concessions so Jeremy Corbyn could claim victory in the talks and sell the deal to his MPs.
Mr Gove, one of the leaders of the Leave campaign in the 2016 referendum, said it would be better to accept the “unpalatable” outcome of a deal with Labour than the “disastrous” outcome of Brexit not happening at all, according to The Daily Telegraph.
Meanwhile, at PMQs Mr Corbyn attacked the PM for 'completely failing' to tackle "burning injustices" in the UK.
His comments came following a Social Mobility Commission report, which warned inequality will remain entrenched in Britain "from birth to work" without urgent Government action.
Touching on Thursday's local elections, Mr Corbyn said: "For many people this Government has delivered nothing but failure."
He recalled Mrs May's pledge to fight against "burning social injustices" on her first day in office, but then flagged the commission's report.
Mr Corbyn asked: "Can the Prime Minister now admit that her Government has completely failed to take action to tackle the burning injustices?"
Mrs May said the commission's chairwoman, Dame Martina Milburn, had highlighted a "real commitment in Government to try to make a difference in this area", before telling MPs: "I want everyone to have the opportunity to reach their potential, whatever their background."
With local elections due to take place around the country issues other than Brexit were on the agenda with both parties looking to make up ground.
Despite this, outside the Commons Mrs May has been unable to escape calls from Labour to move toward a customs union as a way to find compromise on Brexit.
Ms Long-Bailey indicated that movement by ministers on the issue – which Labour supports but Theresa May has consistently opposed – may be key to a solution.
When asked by ITV News if she believed the government would move toward a customs union she said Labour was "hopeful".
"We're waiting to see where they're going to move on all their red lines quite frankly and we're hopeful that they will."
She added: "As yet we've not seen any real movement but we're hopeful that the positive discussions that we've had over the last few weeks will bare fruit and that we will see some real declarations of movement."
She told the BBC Radio 4 Today programme: "I think pragmatically that they potentially may have no option in order to be able to push this deal through."
She added that Labour was now waiting to see how much ground the Government was prepared to give before deciding whether it could make concessions of its own.
“We are fleshing out the details to see how far the Government can move towards us and then we will be able to ascertain how far we are able to move towards them,” she said.
“There are certain issues that we think they will be prepared to move on and we might be prepared to support certain positions.
“There are certain areas which we haven’t seen any movement at all.
“We want to take a view on the whole package, the whole deal, to see if there has been any true movement.”
Her comments came after Mrs May’s de facto deputy David Lidington told the Cabinet on Tuesday that the latest round of discussions had been “serious and constructive”.
The Prime Minister’s official spokesman said further talks were being scheduled “in order to bring the process toward a conclusion”.