The U.K.'s departure from the European Union, long set for March 29, has been delayed until October 31 while Britain's politicians try to break the deadlock.
The new referendum should be a part of any deal with the Conservatives, as Starmer said.
But he said they also agreed it was "imperative" that any exit plan be approved before parliament's summer holiday, which normally begins at the end of July.
Along with measuring European voting intention, YouGov also asked people for their General Election voting intention.
Labour tops the poll, with 24 per cent support, while the Brexit party - led by former Ukip leader Nigel Farage - is second with 20 per cent. Farage's insurgent party is on course to get 34pc of the votes, according to an Opinium poll for the Observer; Labour are on 21pc and the pro-EU Liberal Democrats in third place on 12pc.
He suggested they try to do a deal where the new anti-EU party "runs in the seats that we don't hold", in an interview on the BBC's Newsnight.
The MP for Fareham, who used to chair the hardline European Research Group of Tory MPs, told Labour's Barry Gardiner that "leaving the Customs Union is pretty much what Brexit means".
In a letter to the embattled UK PM, senior Conservatives including Boris Johnson, David Davis and Gavin Williamson said any agreement with Labour over an EU customs union would be "both bad policy and bad politics". The party's official Facebook page doesn't even mention the elections and the prime minister herself won't be out on the stump trying to drum up support. They're in the room negotiating but negotiating to try to make Brexit happen.
Speaking at the rally, Farage was scathing about the government: "It's about a group of career politicians who have never done a deal in their lives, going into this negotiation being outwitted and outfoxed at every turn".
Speaking to Sputnik, the union leader claimed there was now a "very deep, worrying disillusionment with all parties in the United Kingdom parliament and their failure to implement the referendum result", something that would likely impact significantly on the Conservative party in the upcoming elections to the European Parliament.
"I don't think in the real world there's going to be a general election".