Mr Khan – a leading critic of Boris Johnson’s “hard” Brexit deal – has previously called for Britain to consider signing up again to the bloc’s customs union and economic single market.
Sir Keir continues to insist there is no way a Labour government would rejoin the single market or customs union, and ruled out a deal to re-establish freedom of movement as a “red line” he would not cross.
Speaking at a Q&A event at the Fabian Society conference in London on Saturday, the mayor said it was time to have “the conversation” about meaningful alignment.
He said rejoining the EU’s economic alliance should be on the table when the current Brexit deal comes up for review in 2025.
“I’m not saying today we should rejoin the European Union. What I’m saying is that [shadow foreign secretary] David Lammy is onto something when he’s saying we should have a closer relationship with the European Union.”
Mr Khan then criticised the “omerta” – a mafia term for a pact of silence – which means “you can’t talk about being close to the European Union, joining the single market, joining the customs union”.
The London mayor said “the economic cost of this extreme hard Brexit is huge”, having commissioned a recent study which found that it has already cost the UK economy £140bn.
The damning study by Cambridge Econometrics also found that Britain’s decision to leave the EU is set to leave Britain’s economy £311bn worse off by the middle of the next decade.
Mr Khan also said Rishi Sunak’s government is using its Rwanda deportation plan “in the same way they used Brexit – as a deflection of all the problems they’ve caused to our country”.
He called for the Labour party to “speak up” on the benefits of immigration and diversity, rather than try to match the Tories on promises to lower migration levels.
Meanwhile, Labour’s shadow Northern Ireland secretary Hilary Benn claimed that the EU will breathe an “audible sigh of relief” if the party wins the general election later this year.
Mr Benn touted the benefits of a stronger alliance with the bloc, highlighting both economic and security benefits amid growing geopolitical uncertainty.
“The EU is a bit tired of negotiating with Britain,” Mr Benn said in a speech at the left-wing think tank’s event. “And there is no doubt that the arrival of a Labour government, if we win ... will be greeted in the EU with a big and audible sigh of relief.
“And I think eventually there is going to be a moment when we look at each other across the Channel and say, ‘You know what, we’re still big and important trading partners, we’re still friends and allies, we are cooperating on foreign policy and security and we need to do more of that ... so wouldn’t it make sense to have a closer relationship?’”
Mr Benn added: “And why does this matter so much? Well, clearly it matters economically but there’s something else that is even more important, and that is building alliances in a dangerous world.”
The Labour frontbencher pointed to conflicts in Ukraine, the Middle East and Sudan, tensions over Taiwan, the threat of climate change and the risk of a second Donald Trump presidency.
“We should come together with those who share our common values to defend those values in the face of these threats,” he concluded. “The best way to protect your sovereignty is to work with others.”
Sir Keir has promised to seek a new veterinary agreement with the EU to align some stands in a bid to ease current trade friction. The Labour leader has also made vague pledges to examine the current Brexit trade deal when it comes up for review in 2025 with a view to making it “better”.