And he said Mr McCluskey should "stop targeting me" and argue for a "People's Vote" on Brexit.
Jeremy Corbyn has said he will be "bound" by the decision of party members in a vote on Tuesday.
He added: 'We're respecting the referendum.
Corbyn's second-in-command, finance spokesman John McDonnell, was clear - the question should not involve a re-run of the 2016 referendum by including an option to remain in the EU.
The Conservatives seized on Labour's potential support for a people's vote, with the prime minister tweeting that the party wanted to "take Britain back to square one - betraying all those who voted in the 2016 European Union referendum".
"We need to be very wary of being labelled as a middle-class pro-Remain or pro-EU party", Barker shouted at hundreds of Labour supporters amassed on Liverpool's pier head on Saturday evening to see Corbyn take the stage.
"Get behind the party that made you, and join the rest of this movement in fighting to get rid of this rotten government, to give our country the fresh start it needs and make Jeremy Corbyn the people's prime minister".
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Delegates at the party's annual conference in Liverpool will consider a motion Tuesday saying "all options" should remain on the table, "including campaigning for a public vote".
'All the polling that we have seen is that the country is still pretty split down the middle'.
Last night Sir Keir said: 'If Theresa May brings back a deal that fails our tests - and that looks increasingly likely - Labour will vote against it.
"Dozens of motions were submitted to (conference), thousands have marched on the streets, and millions more have called for a People's Vote", Labour lawmaker David Lammy, who has a London constituency, said on Twitter.
The Labour leadership has been under intense pressure to shift its position on a second referendum, led by a cabal of Remain-supporting MPs and, crucially, a growing number of unions. If that does not succeed, then the party should keep all options on the table, including the possibility of a second referendum on a question yet to be determined.
"If the government is confident in negotiating a deal that working people, our economy and communities will benefit from, they should not be afraid to put that deal to the public".
He added: "We've now got to turn up the noise until we secure a public vote which gives us the option to remain in the United Kingdom". They've spent two years backing the Tories' catastrophic Brexit and are even now trying to ignore their own members who want a People's Vote with the option to remain in the EU.