Labour's deputy leader Tom Watson has described another Brexit referendum as "the least worst option" and urged his party to throw its weight behind one.
Speaking to the BBC, he said Labour should then fight for Remain, even though "we might lose some votes".
Jeremy Corbyn has resisted calls to fully endorse another public vote, only calling for it in some circumstances.
But Mr Watson said Labour would pay "a very high electoral price" if it did not have "a clear position" on Brexit.
The nuanced position was blamed for Labour's performance at the European elections - it came third behind The Brexit Party and the Liberal Democrats, with its share of the vote falling to 14%.
Afterwards, several senior figures criticised a lack of clarity on Brexit, and last week, MPs expressed their frustration at a heated meeting of the Parliamentary Labour Party.
The PLP is still split, though, with some MPs in Leave-supporting areas warning against backing a further public vote.
The shadow cabinet was due to meet on Monday to discuss Brexit, but the meeting has been postponed.
Mr Watson - who has repeatedly put pressure on Mr Corbyn to back a further referendum - told the BBC's political editor Laura Kuenssberg he believed it was now the only choice available.
Theresa May's Brexit deal with the EU has been rejected by Parliament three times and the UK currently has until 31 October to come up with another way to leave.
"Sometimes in politics your choices are the least worst option," Mr Watson said. "It is my honestly held view that Parliament will not be able to get a deal on Brexit and therefore the only choice, reluctantly, is to ask the people to take another look at it."
When asked if he would leave the Labour Party if things did not change, he replied, "I'm never going to leave the Labour Party," but added "sometimes I wonder whether the Labour Party is leaving me."
Earlier, in a speech to the Centre for European Reform, the deputy leader said Labour must be honest about the EU's strengths.
"Pro-European is who we are and who we have always been. Our members are Remain. Our values are Remain. Our hearts are Remain."
He told the BBC Labour "might lost some votes if we change position", but added: "I think it's incumbent on us to give an honest account of ourselves and make the case for why we've changed our position."
Mr Watson is calling for a one-off meeting or ballot of members to be held to vote on a shift in policy - warning Labour could not afford to wait until its party conference in late September.
But as he gave his speech, Labour chairman Ian Lavery - who is against another referendum - tweeted that "ignoring Leave voters" was not a sensible move.
Brexit has turned this country into a toxic nation.— Ian Lavery MP (@IanLaveryMP) June 17, 2019
However ignoring the 17.4m leave voters isn’t politically smart nor indeed particularly democratic.
Labour MP John Mann warned adopting an overtly Remain position would lead to Labour losing the next general election "by a significant amount".
He said if Labour "turned its back" on voters in the North who voted Leave, "then Tom Watson won't be deputy, Jeremy Corbyn won't be prime minister."
Labour MP Kerry McCarthy said she would commend Mr Watson for "speaking out", but shadow ministers needed to "meet urgently for a proper discussion on Brexit".
"We need to be clear where Labour stands, and if [the] shadow cabinet can't agree, put it to the members," Ms McCarthy posted on Twitter.
Mr Watson has received support from a number of colleagues, including Jess Phillips and Anna Turley.
Another MP, Siobhain McDonagh, tweeted: "I have had my differences with Tom Watson over the years but this video is brilliant and his argument is bang on! So many Labour members will be cheering him on!"
Laura Kuenssberg says plenty of Labour MPs are worried because they represent constituencies with Leave voters, but there is no question the balance in the party is on the other side.
"There are plenty of senior people - including those absolutely loyal to Jeremy Corbyn - who think it is time for the leadership to make a clearer statement arguing for another referendum and for Britain to stay in EU," she says.
"Some of those think it is vital to do before the summer and they predict we may end up with an election in the autumn with the Tories arguing for Leave and Labour arguing for Remain."
However, Mr Watson said all strands of opinion within the party are entitled to be heard.
He also argued that the "core" EU values of internationalism, solidarity and freedom are also the values of Labour.
"Some people have begun to equate support for Europe with class identity - I don't think that's right or helpful," he said.
"The majority of Labour people are supportive of Europe and that support is not dictated by social class."