In a speech in London on Wednesday, Mr Watson – already a hate figure for many allies of Labour leader Jeremy Corbyn – will demand a referendum before a general election.
Meanwhile, at the Trades Union Congress (TUC) conference in Brighton, Sir Keir will also argue a public vote is the only way to overcome the Brexit impasse.
In what looks like a co-ordinated move with Mr Watson, he will say: “We will have to break the deadlock.
“And clean up the mess left by the Tories. A referendum is the only way to do so.
“And that is why Jeremy was right to say at Congress yesterday that an incoming Labour government will commit to a referendum.”
In his own speech in Brighton on Tuesday, Mr Corbyn said Labour would commit to a public vote during the next general election campaign “with a credible option to Leave and the option to Remain”.
While Sir Keir is careful to back Mr Corbyn, Mr Watson’s demand will be seen by his critics in the party and Corbyn allies as a challenge to the Labour leader’s authority.