ITV News video report by ITV News political correspondent Daniel Hewitt
Mr Corbyn has hinted the Labour Party may move closer to backing a second referendum after the party received a drubbing which saw it lose half its MEPs and take just 15% of the vote.
Shadow chancellor John McDonnell said Labour could unite the party and country by “taking (the) issue back to people in a public vote”.
But shadow justice secretary Richard Burgon insisted Labour had the right approach in seeking to appeal to both Leavers and Remainers – even though it was not suitable for the European elections.
Shadow foreign secretary Emily Thornberry and deputy Labour leader Tom Watson both called for a more strident position from the party on the issue of a second referendum.
In a statement, the Labour leader said the EU elections had become “a proxy second referendum”.
He said: “With the Conservatives disintegrating and unable to govern, and Parliament deadlocked, this issue will have to go back to the people, whether through a general election or a public vote.”
He added: “Over the coming days, we will have conversations across our party and movement, and reflect on these results on both sides of the Brexit divide.”
Labour won just 10 MEPs in England and Wales and although Scotland’s result is not yet official, Mr Corbyn’s party is on course to lose its two MEPs there. In 2014 the party won 20 MEPs across the UK.
Mr Corbyn’s party trailed in third place, behind the Brexit Party and Liberal Democrats nationally.
Wigan MP Lisa Nandy told ITV News voters are tired of the party's indecision over Brexit, adding those advocating for a second referendum should first try to implement the result of the original poll.
But Mr Burgon said: “I think the message of trying to bring people together who voted Remain and Leave is the right message.
“It was never going to work in this kind of low-turnout EU election where the people most interested in this important issue of Brexit, whether it is to Remain or Leave, came out to vote. A general election would be very different.”
He told ITV’s Good Morning Britain that Labour could support a referendum to stop a new Tory leader taking the UK out of the EU without a deal.
“Whether it be a no confidence vote in the Government, a general election or a public vote, we will use whatever mechanism necessary to stop a disastrous no-deal Brexit,” he said.
Mr McDonnell said Labour could not hide from the “hit” it took in the elections.
Faced with the prospect of a “Brexiteer extremist” taking over from Theresa May, a public vote would “unite our party and country”.
Mr Watson said the party cannot go into a general election without a clear position on a second referendum and a special conference may be needed to decide a Brexit policy.
He tweeted a link to a survey on his website, asking members how the party should agree a new Brexit policy.
He said: “Following the disastrous EU election results, Labour urgently needs to re-think its Brexit position and realign with members and voters.”
Ms Thornberry – who was Labour’s representative on the BBC’s election programme – tore into her party’s position live on air.
She said: “We went into an election where the most important issue was ‘what was our view on leaving the European Union?’ and we were not clear about it.
“We were not clear on the one single thing that people wanted to hear and that wasn’t their fault.
“We sent people out to campaign on that and, unfortunately, we just weren’t clear enough.”
She said: “I fear we will have no deal and we must be clear it will be a disaster for the country so we must have a second referendum.”
Tottenham MP David Lammy accused Labour of “hiding” on the Brexit issue but now needed to campaign to remain in the EU.
“Labour tried to ride two horses in this race,” he said. “We fell flat on our faces and got trampled.”