According to a report in The Observer at the weekend, referenced by the Liberal Democrat leader Sir Ed Davey at PMQs, those in government pushing for the UK to rejoin the customs union include Baroness Shafik, Sir Keir's chief economics adviser.
Others admit privately that they see rejoining the customs union as the simplest way to inject growth into the British economy. This would mean that the UK applies the same tariffs as EU members to goods imported from countries around the world. There are no tariffs applied to trade between countries inside the customs union.
One government source involved in this issue said that even if Sir Keir wanted to rejoin the customs union, it would not be straightforward. They said the EU would, initially at least, be likely to make unpalatable demands of the UK in exchange - pointing to the recent collapse of talks over the UK joining an EU defence fund.
Moreover, joining the customs union would mean the government giving up on its independent trade policy, and therefore achievements that the prime minister himself talks up regularly, including the free trade deal with India and the deal to reduce tariffs with the US.
In any case, would rejoining the customs union be politically possible?
Well, surprising people in the Labour fold think the answer to that is yes. At the general election, Labour was eager to reassure those who backed Brexit in 2016 that it would not reopen the European question. The party won back many seats it had lost in 2019.
Yet one MP who represents a 'red wall' seat which voted strongly to leave the EU said: "It's much less visceral than it was. Even in constituencies like mine there's space to go quite far I think. The single market is a non-starter because you reopen the immigration issue.
"But there's space to rejoin the customs union. It's about trade. No-one is going to object to better trade."
A cabinet minister agreed: "Voters just don't really care about Brexit any more."
Some in Labour believe that those voters who do still care about Brexit include progressives who the party risks losing at the next general election to the Greens or the Liberal Democrats.
They see a significantly closer relationship with the EU as a way of securing those voters in Labour's coalition, as well as drawing a sharp political dividing line with Reform UK and the Conservatives.
And polls suggest a majority of the public now believes the UK was wrong to leave the EU.
The cabinet minister said: "Brexit is becoming like the Iraq War. Lots of people supported it at the time but now you can't find anyone who admits to it."