Britain should work towards rejoining the EU in the next decade, Tony Blair has said, adding to growing pressure on Andy Burnham to abandon Labour’s reluctance to revisit the 2016 referendum.

The former prime minister has urged Burnham to use his 10-year plan in November to put the country’s membership of the union back on the table, a decade after it voted to leave.

In a paper entitled A Path to Rejoining a Reformed Europe, the Tony Blair Institute (TBI) calls for the UK to accept free movement of people and make a financial contribution as the price of entry.

The intervention comes after Burnham declined to rule out promising to rejoin the EU in the next Labour manifesto, though it also threatens to reopen the deep rifts caused by the vote 10 years ago.

In a foreword to the paper, Blair argues: “Britain should signal an intention if the terms are right to enter into a structured formal relationship with Europe and initiate a process of negotiation where everything is on the table but nothing agreed until everything is agreed.”

TBI researchers say in the paper itself: “Over the next decade, Britain should work towards the ambition of rejoining a significantly reformed EU.”

Blair is the latest in a series of Labour figures to back rejoining the EU. Wes Streeting, the defence secretary, has called the Brexit vote a “catastrophic mistake” and said the party should aim to rejoin the bloc, while Burnham himself told the Guardian last year: “I hope in my lifetime I see this country rejoin the European Union.”

But Burnham has said he does not wish to pursue membership of the union in the near term. When asked by reporters this week if it could be in the next election manifesto, he said it was a question “for another day”.

Others are sceptical of the electoral wisdom of giving the appearance that the party wants to undo the result of the 2016 referendum, especially in areas of the country where Reform UK has made inroads in recent years.

Lisa Nandy, the culture secretary, dismissed the idea earlier this year as she warned against “reopening the circular arguments that we ended up in as a country”.

However, pollsters suggest Labour could benefit electorally by promising to rejoin. A new megapoll conducted by YouGov on behalf of the pro-EU campaign group Best for Britain suggests the party could more than double its seat total if it goes into the next election promising to rejoin.

The survey of more than 13,000 people, undertaken in June before Burnham became prime minister, found that a pledge to seek membership would help unite progressive voters behind Labour while doing nothing to boost Reform’s vote.

The results suggested the move could take Labour from a projected 136 seats to 321 – though those totals will have changed since Burnham came to office, which has given the party a poll lead for the first time in months.

The TBI paper calls for Britain to try to accelerate European market reforms using possible re-entry as leverage. It argues that pro-reformers in Brussels would have a significant incentive to work with Britain on a project to make the EU more economically liberal and to reform its regulations on energy and technology.

It adds that Burnham should scrap the government’s red lines of not rejoining, not allowing free movement and not entering the single market or a customs union. The paper argues that the prime minister should replace them with a single promise only to sign up to rules and initiatives that prove of economic or strategic benefit to the UK.

TBI has advocated in recent years for the UK to slow down its push to achieve net zero carbon emissions and is a heavy backer of increased use of artificial intelligence in government.

Its policy positions have brought scrutiny on some of the institute’s most high-profile funders, who include the technology billionaire Larry Ellison, and its clients, which include the government of Saudi Arabia.