“It is conceivable that there is no majority position for any deal in the current arrangements in parliament,” he said.
“So, even though we’d prefer a meaningful vote in parliament – and I have said many, many times it’s highly, highly, highly unlikely we will support a people’s vote, and we have not called for it – we haven’t taken that off the table.
“Because there are a rare set of circumstances where parliament just can’t make a decision on it and, in those circumstances, you’d keep your options open.”
Mr Watson, speaking on BBC Radio 4’s Today programme, repeated that it was “highly, highly unlikely we’ll go down that route”.
Nevertheless, the anti-Brexit Best for Britain group seized on the comments as further evidence that the prospect of a “people’s vote” was growing.
“His words add to a growing chorus of people disillusioned with Brexit who want to take back control of a process trapped in the political channels of government and parliament,” said Eloise Todd, its chief executive.
Jacob Rees-Mogg, the Tories’ most influential back-bench Brexiteer has said he would join with Labour in voting against the “defeatist” plans – suggesting other Tory Eurosceptic MPs would do the same.
It raises the prospect of Brexiteer Tories joining forces with opposition parties to throw out the deal – even if the EU agrees to it, which still seems unlikely.
Mr Watson, speaking on BBC Radio 4’s Today programme, suggested Labour would oppose the Chequers plan, agreeing with Mr Davis that it appeared a non-starter.
On the “facilitated customs arrangement”, he said: “We think it’s very complex and requires a technical solution that doesn’t exist – so, in effect, that’s unworkable.”
Other huge problems in the Chequers plan, for Labour, include no EU trade deal for Britain’s huge services industry
Mr Watson said the stark Tory splits “electorally might help my party”, but insisted: “We want to work with the government for best deal.”
The deputy leader also revealed that Labour was “not decided how to vote next week” when pro-EU Tories had been expected to rebel to try to keep the UK in the customs union.
That revolt now looks unlikely, after Anna Soubry, Nicky Morgan and Stephen Hammond heaped praise on the prime minister for her Chequers proposals – even though they would mean leaving the customs union.