In a dramatic U-turn it called for the UK to remain signed up to Brussels for up to four years after departure in 2019.
That would include continued free movement for EU migrants.
Brexit spokesman Sir Keir Starmer is also understood to want a future Labour government to strike a deal that could extend single market membership indefinitely.
His policy shift, which was signed off by the Labour leader’s office last week, reversed the party’s previous promise to back a clean break from the EU.
Richard Tice, of the pressure group Leave Means Leave, said: “This is one of the most astonishing U-turns in modern political history.
By backsliding on Brexit, the Labour Party has completely betrayed its manifesto commitment and the people who voted for it less than three months ago in the General Election.
“To remain in the single market and customs union during an unlimited transition period would create long-term uncertainty and be disastrous for British businesses.”
They would be “unable to reap the economic benefits of Brexit until it is fully implemented, not to mention the weak negotiating position it would put Britain in”.
He said that a further suggestion that Britain could negotiate a new single market and customs union arrangement is “delusional and completely at odds with what the British people voted for”.
A Tory spokesman said: “The truth is Labour have no idea what they want and this is a weak attempt to kick the can down the road.”
Tory MP Maria Caulfield, who was on the Commons EU Exit Committee, said that only last month Labour leader Jeremy Corbyn was stating that a Labour government would leave the single market.
“Now they have completely U-turned and are out of touch with the wishes of the British people by campaigning to stay in the single market and the customs union.”
Former Ukip leader Nigel Farage said Corbyn “has now betrayed every Labour voter at the General Election”.
Sir Keir, the shadow EU exit secretary, said: “Labour would seek a transitional deal that maintains the same basic terms that we currently enjoy with the EU.
“That means we would seek to remain in a customs union with the EU and within the single market during this period.”
His proposal would mean Britain retaining an open-door for EU migrants and remaining under the EU Court of Justice.
It would also prevent Britain signing new trade deals.
It would also pay billions to Brussels budget each year.
Labour sources also indicated their chiefs wanted the option of staying in the single market and customs union for ever if the EU could agree a deal on immigration.
The shift will mean the Prime Minister could struggle to push Brexit legislation through if Tory MPs rebel and vote with Labour.
Kate Hoey, a pro-Brexit Labour MP, said: “Sir Keir is tearing up a key and direct promise made in the manifesto. I would not be surprised if all those Ukip voters who came back to us abandon us once more.”
August 28th, 2017: Express