Senior Downing Street officials have been reaching out to thinktanks and pollsters as they look to test public support for the UK rejoining the EU.

Aides to Andy Burnham have been talking to EU experts in a range of organisations about how the public would view a promise to rejoin, as momentum builds behind including such a pledge in the next Labour manifesto.

Some in No 10 are keen to promise that Labour would at least negotiate over re-entry to the bloc, something Burnham has previously said he wants to see happen "in his lifetime". Others are nervous, however, that making such a promise could revive the flagging political fortunes of Nigel Farage, the Reform UK leader.

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

The prime minister told delegates on Tuesday: "Brexit has done more harm than good. So I will take the opportunity around the [UK-EU] summit to lay out to you, to lay out to the country, what I believe are the different options for Britain's long-term relationship with our European partners."

On Wednesday he clarified that this would include an option to rejoin – something ruled out by his predecessor Keir Starmer.

Burnham told BBC Radio 4's Today programme: "We could stay as we are. That's definitely an option, if people think this is the right place to stay. We could look at what [the former Conservative chancellor] George Osborne has said about a customs union, we could look at what the Liberal Democrats said at their conference about the single market, or we could go all the way."

Burnham's comments sparked joy among pro-EU campaigners, who argue that rejoining is the best way to boost UK growth rates – and outrage from political opponents.

Alex Burghart, the Conservative deputy leader, said: "Labour cannot blame a democratic vote from over a decade ago for the consequences of their own economic decisions."

The Guardian has learned that Burnham's comments are part of a wider effort by No 10 to test consensus over rejoining, with a view to possibly including such a promise in the next manifesto.

Officials have been in talks with several prominent thinktanks and pollsters, who have been working on projects to flesh out arguments over rejoining and test public opinion.

Last week, the Tony Blair Institute released a paper arguing that Burnham should seek talks with Brussels over becoming a member once more but should only do so if Europe relaxed its regulations in sectors including energy and technology.

On the same day, the pro-EU campaign group Best for Britain released a megapoll conducted by YouGov suggesting Labour could more than double its seat total if it went into the next election promising to rejoin.

Over the coming weeks, the influential Institute for Public Policy Research will release a paper calling for the country to rejoin, while Think Labour has also been looking into how difficult it would be to do so.

Some in Downing Street are buoyed up by polling which shows that promising to re-enter the bloc does little to boost support for Reform. They also point out that Farage attacked the prime minister's speech for the promise to change the pensions triple lock, rather than his comments about Europe.

The Guardian has been told that some at the top of government wanted the prime minister to use this week's speech to promise to begin work on rejoining after the next election, rather than carrying out a review.

In the end, such an option was ruled out because the party was already bracing itself for a fight over pensions and did not want to give succour to Farage while he faces accusations about millions of pounds' worth of donations to his party.

No 10 said the prime minister did not ever consider making such a promise in the speech and that it was not included in any draft of it.

Some believe it would be difficult to persuade Brussels, which has engaged in multiple rounds of talks about Britain's relationship with the EU, to embark on negotiations over re-accession.

But in a major boost to supporters of rejoining, Emmanuel Macron on Wednesday welcomed Burnham's comments. "Welcome back," the French president said in English at a press conference in Madrid on Wednesday.

He continued in French: "If the British want to come back and if the prime minister is moving in that direction, I think that is very good news, both for his country and for the Europeans."

An EU spokesperson said: "The UK is a close and valued partner of the European Union. It is for the UK government to set out how it wishes to develop its relationship with the EU going forward. We remain open to engaging on the options it may put forward."

Burnham's willingness to look at all options seems to have been accompanied by fresh engagement on the EU's Made in Europe legislation, which threatens to lock UK goods out of certain supply chains on the continent.

No 10 has sought to have British manufacturers classified as European for the purposes of that legislation, with officials threatening to delay an EU-UK summit until London receives signals that there could be an accommodation.

Since Burnham's speech, however, there is now talk of holding the summit on 20 November.

One EU diplomat said it was necessary to get Starmer's old reset agenda concluded before talks of "a real reset" could begin, confirming support for a UK-EU summit as early as possible.

A second diplomat said there was "a good feeling" that Burnham was now looking at all options and seeking a consensus, "something Brussels has wanted to see".