Senior party figures have confirmed the move is likely and that Labour could change its position by March or April.
If Britain remains inside the customs union, it will not be able to negotiate its own trade deals after Brexit, and will have to leave this to the EU.
However Labour hopes the customs union can be reformed, so Britain can have a seat at the table in future trade negotiations.
There are still significant obstacles to the policy being adopted, however senior Labour insiders, including MPs in Brexit-backing seats, believe it could be announced in the coming months, according to the Times.
The change could put Remain-supporting Conservatives in a difficult position. Many of these support remaining in the single market and customs union but would have qualms about voting against the government.
A significant defeat for Theresa May’s Government as Brexit negotiations continue could bring her down as Prime Minister.
However Labour’s support for customs union membership is currently being blocked by shadow trade secretary Barry Gardiner.
It also faces opposition from left-wing MPs close to Jeremy Corbyn, who “don’t want to be dragged by backbench Blairites”, although they are not in principle opposed to the policy.
Shadow work and pensions secretary Debbie Abrahams faced a grilling from Daily Politics presenter Andrew Neil as news of the possible change came to light.
She admitted the issue had been discussed in shadow cabinet on Tuesday, but revealed Labour had not discussed proposing a concrete model for a new customs union.
Ms Abrahams said: “We have said a customs union, we are not hung up about the customs union. We’re not hung up about the model. What we are focussing on is a negotiated settlement that will have benefits and protects jobs and the economy.”