Jeremy Corbyn, triumphant after defying expectations in the general election, has said parliament should hold a vote to secure the future of EU nationals living in Britain.
The Labour leader told the BBC that it would be "good gesture" from parliament "to vote now to agree that all EU nationals can remain in Britain."
Brexit negotiations are due to commence in 11 days, on 19 June. German MEP Manfred Weber, who leads the largest group in the European Parliament, said the planned start date of the Brexit talks was "uncertain" after the shocking election result.
He said that Theresa May had "brought chaos" rather than stability to the UK.
"Our position is clear: we want good cooperation with the UK, but Brexit means leaving the EU with all its benefits," Weber said on Twitter.
But Corbyn insisted that the Brexit negotiations would go ahead and that whoever is in office in 11 days time would have to conduct the talks.
He said the Labour Party would fight for a "jobs first Brexit" and a strong trade deal with Europe.
"We are ready to serve this country. This is what we fought this election for," he said.
Europe's Chief Brexit negotiator, Michael Barnier, said that the Brexit talks should only commence "when UK is ready" and the "timetable and EU positions are clear."
"Let's put our minds together on striking a deal," he wrote on Twitter.
Corbyn repeated his call for the prime minister to resign.
"She fought the election on the basis that it was her campaign; it was her decision to call the election, it was her name out there; she was saying she was doing it to bring about strong and stable government," he said.
"Well, this morning, it doesn't look like a strong government, it doesn't look like a stable government, it doesn't look like a government that has any programme whatsoever."
Conservative MPs say that Theresa May has no intention of standing down and is resolute on forming a government, most likely by making a pact with Northern Ireland's Democratic Unionist Party (DUP), which won 10 seats.