Andy Burnham’s personal preference to rejoin the European Union after the next election is supported by almost half of all voters, a new megapoll has found, although cabinet ministers concerned about reopening Brexit divides are privately urging him to proceed with caution.

The poll by the Messina Group (TMG) found that renegotiating the UK’s relationship with the EU could boost Labour’s vote share by as much as seven points, if the prime minister offered a fresh referendum in which he campaigned to re-enter the bloc.

Almost a third of voters said an in-out referendum would be needed first, while more than one in six felt that a general election mandate would be sufficient, and the same proportion thought parliament could make the decision.

The overall findings will bolster the case of those inside Downing Street who want to promise at the next election that Labour would at least negotiate over re-entry, which the prime minister has said he wants to see “in his lifetime”.

Burnham used his first Labour conference speech as leader on Tuesday to announce a review of the options for Britain’s future relationship with Europe, and subsequently added that these should include rejoining entirely.

However, there are nerves among some Labour MPs, especially those with constituencies that voted for Brexit that even testing out public support for the move would be fraught with political risk and a potential boost to Reform.

Two-thirds of cabinet ministers, including Burnham himself, his first secretary, Louise Haigh, chancellor, John Healey, and foreign secretary, Ed Miliband, represent seats that backed Brexit, and previously said they would respect the 2016 vote. However, public opinion has shifted.

Before Burnham’s speech, several ministers had publicly cautioned against the move. Lucy Powell, Labour’s deputy leader, said while the defence and security relationship could be better, going beyond EU reset plans inherited from Keir Starmer was “not possible nor desirable”.

The trade secretary, Jonathan Reynolds, said “constitutional questions” around Britain’s EU relationship were causing uncertainty for business. The home secretary, Shabana Mahmood, said rejoining was “not a simple, straightforward thing” and the costs might be unacceptable.

Downing Street’s strategy appears to be to put the focus on those who made big promises on Brexit but then failed to deliver, rather than accusing those who voted to leave the EU of making the wrong decision. But the government’s shift in position has caused some anxiety among Burnham’s team.

One cabinet minister told the Guardian: “Brexit clearly hasn’t worked and the promises made haven’t been kept so Andy is right to open up the question of what the future relationship looks like. But he definitely needs to proceed with caution because we don’t want to restoke division”.

A second said: “Andy is right to say we need some clarity around our long-term relationship. Security and trade cooperation are higher up the agenda than they were 10 years ago. But whether that adds up to a real desire for big change in the population I don’t know. My constituents will be cautious.”

A third added: “There are obviously different views across cabinet about it, but the Made in Europe agenda means we do need to talk to the EU about what relationship they want with us long term, as what they are currently proposing is very challenging.”

Another cabinet source said: “There’s an awareness among a number of cabinet ministers with leave voting seats that opinions are going to differ and the arguments about the future will have to be carefully made about the national interest.”

A fifth said there were genuine questions about the best option on the table. “I’m sceptical if we go back in, about the terms we’d be going back in on, and whether that can really resolve the problems in people’s lives. But if you look at the global context you cannot start to solve things for people without having a closer relationship with the EU.”

TMG polled 11,461 people, weighted by referendum result, from 12-24 September on seven different options for relationships with the EU. The most popular was rejoin, with 49% in favour and 28.5% against. The second was to keep the existing relationship but cut as many trade barriers as possible – as the government is attempting to do with its reset – and which was supported by 48.2%, with 20.9% against. A customs union came next, on 47.1% versus 20.7%. Joining the single market and associate membership were virtually neck and neck – 45.1% and 45.5% of respondents respectively support them – while sticking with the existing relationship was backed by 42%.

The seventh option – the UK moving further away from the EU – was supported by only 27.9%, with 41.2% against. A plurality of voters – 25.9% – said they wanted the UK to rejoin the EU immediately, while a slightly lower proportion – 22.9% – said it should happen in the next three years, and 8.4% over the next decade, while 23% said they did not want a closer relationship at all.

TMG also tested political messaging for each of the options to see how they could affect Labour’s vote share in an election, and found that it went up – by between 4.3 and 7.4 percentage points – for all the scenarios that resulted in closer ties, with most votes coming from the Greens and Lib Dems.

The polling showed that more than two-thirds of voters thought that trading with the EU was more important than the US or China, in the event the UK had to choose; four in five people want to keep the pound; and a majority want to stay in the ECHR – 53.4% versus 24.2% against. However, it also found that Brexit arguments over immigration control and sovereignty still resonated.

Senior Downing Street officials have been in contact with thinktanks and pollsters as they look to test public support for the UK rejoining the EU. The TMG research is expected to be shared with the current operation.

Some inside No 10 are nervous, however, that promising to renegotiate entry into the bloc could revive the flagging political fortunes of Nigel Farage, who said on Thursday Burnham had given him “the biggest boost in my career” by reviving the Brexit debate The TMG research found that 29.7% people thought the UK had been right to leave the EU, while 52.4% said it had been wrong, and 17.9% didn’t know.

On whether it had been a success or failure, 12.2% said a success, 54.9% a failure, 23.2% said neither and 9.7% did not know.

Ben Judah, a visiting fellow at Chatham House who was an adviser to David Lammy as foreign secretary, told the Guardian: “These polls are a massive validation of Andy Burnham’s boldness on Europe. There is close to a national consensus that Brexit was not a success.

“Against this background these results show that Europe as an issue has a unique ability to unite the left and grow the Labour vote share among swing voters. The bolder the stance towards rejoin, the bigger Labour’s reward.”