by Jennifer McKiernan, Political reporter and Laura Kuenssberg, Sunday with Laura Kuenssberg
The prime minister told the BBC’s Laura Kuenssberg it would be "better looking to the single market rather than the customs union for our further alignment", in order to protect trade deals with India and the US.
But he ruled out revisiting manifesto promises not to rejoin the EU single market or customs union, or to end freedom of movement.
The comments are the clearest indication yet that Sir Keir wants to pursue a closer relationship with Europe in a broader number of areas.
The Conservatives said the PM was using Brexit as an excuse for the UK’s economic struggles.
The UK is already lining up with Brussels on some rules around food and agriculture to allow access to the economic European trading zone known as the single market.
Sir Keir told Laura Kuenssberg: "I think we should get closer, and if it’s in our national interest to have even closer alignment with the single market, then we should consider that, we should go that far.
"I think it’s in our national interest to go further."
He added: "I actually think that now we’ve done deals with the US, which are in our national interest, now we’ve done deals with India which are in our national interest, we are better looking to the single market rather than the customs union for our further alignment. And it wouldn’t be in our interest now to give up."
Stressing that building closer economic ties was a "sovereign decision", the prime minister said that doing so had led to the best relationship with the EU "for 10 years".
"What I’m saying is, there are other areas where we should consider whether it’s in our interest to do the same and align with the single market," he said.
"Now, that needs to be considered on an issue-by-issue, sector-by-sector basis, but we’ve already done it with food and agriculture and that will be implemented this year."
Sir Keir’s comments follow pressure from within the Labour movement to go further on a customs union, with 13 backbenchers backing proposals that would pave the way for such an arrangement in a Commons vote in December.
Reminded of his promise that "Brexit is safe in my hands", the prime minster insisted this was not an attempt to reverse it - because nobody wanted to be "picking over the bones of Brexit" but rather "looking forward" at what was in the national interest.
Elsewhere, concerns have been raised that current negotiations around a youth mobility scheme for British and EU students could end up requiring the UK to sign up to the free movement of EU citizens.
Sir Keir said: "That is not the return to freedom of movement, we’re not going back to freedom of movement.
"But I personally think that young people having that opportunity is a very good thing.
"And when we announced just recently that we’re going back to the Erasmus scheme so that young people can study, exchange, research can be done in a much better way than it is now - that was widely welcomed."
Conservative shadow foreign secretary Priti Patel said Labour’s "Brexit betrayal" was becoming clearer, and said Sir Keir would be "surrendering our freedom to cut regulation and strike our own trade deals" by pursuing closer alignment with the single market.
Speaking to BBC News, she also said: "He’s unpicking and unravelling Brexit and this is another excuse for him, rather than fixing the fundamental problems that he has created and his government and his chancellor have created in the UK economy."
Appearing on Sunday with Laura Kuenssberg, Liberal Democrat MP Layla Moran welcomed Sir Keir’s "warm language" on EU alignment, and said her party had "long argued this is the direction we should be moving in".
She said the government needed to be "shifting the big dials" on the economy, but said the Lib Dems would prioritise the customs union over the single market, as they felt the UK could not re-enter the latter without allowing free movement.
At the end of last year, the UK’s most senior trade unionist, Paul Nowak, told the BBC "the closest possible economic and political relationship with the European Union" was "essential" to boost economic growth.
Deputy prime minister David Lammy also told the News Agents podcast last month that rejoining the EU’s customs union was "not currently our policy" - but highlighted that Turkey had seen growth as a result of its own cooperation with the bloc.
Turkey’s deal does not apply to agricultural goods or services but removes point of origin checks, which have proved troublesome for the post-Brexit UK, and has been suggested as an alternative for the UK.
By: N. Peter Kramer