Prominent remain supporters including Tony Blair and John Major have been working with Nick Clegg and Peter Mandelson on a diplomatic mission to try to persuade European leaders to stop Brexit.

Clegg, the former deputy prime minister, began the mission independently but has taken on the role of informal shop steward to the grandees.

In recent weeks Clegg has met Mark Rutte, the Dutch prime minister, Wolfgang Schäuble, the German Bundestag president, Peter Altmeier, the German economics minister, Sigmar Gabriel, a former German foreign minister, and senior officials in the foreign policy team of the French president, Emmanuel Macron.

Last week Blair met senior politicians in Germany and Austria and Italy’s new interior minister, Matteo Salvini. The People’s Vote campaign, working alongside Clegg, has appointed Tom Cole, a former EU commission official, to stay in regular contact with EU embassies in London.

Reports from the shuttle diplomacy are fed into the weekly meeting on Wednesday mornings of remain-backing MPs, which is chaired by Chuka Umunna, the Labour MP and lead coordinator on the coming parliamentary Brexit endgame.

One source said: “We are not trying to subvert the government negotiations but we are trying to make sure European leaders are plugged into British politics and are not just getting information from the UK government.”

The greatest difficulty of the alternative diplomatic mission is that it is self-appointed and represents no party.

Speaking from Italy, Clegg told the Guardian: “The aim of the visits is to persuade EU leaders that British politics has made the option of remaining inside the EU viable.”

Arguing that the British discussion about a further referendum was often held in isolation from European politics, he explained: “My primary purpose has been to get European politicians just prepared for the possibility that Britain is not capable of delivering a workable Brexit and they may need to be ready for that.

“It is a question of pressing them to keep open the extension of the article 50 process beyond the March deadline to give UK negotiators more time including to prepare legislation on a people’s vote.”

Clegg is well connected to undertake this shuttle diplomacy, having known the chief Brexit negotiator, Michel Barnier, for nearly 25 years. Some key members of Barnier’s negotiating team worked alongside Clegg when he worked at the commission.

Clegg said: “For a year after the referendum I was received with a mixture of curiosity and pity on the basis it was not remotely likely anything was going to stop Brexit. The atmosphere has changed.”

He pointed to the Austrian chancellor, Sebastian Kurz, saying he preferred a pause in the negotiations rather than a hard Brexit.

Clegg is also trying to gather support for a new offer from Europe in the event of a further referendum. “We cannot just turn the clock back to just before the 2016 referendum. There would have to be some changes of freedom of movement; we cannot just put Humpty Dumpty back together again.”

Blair, who met Barnier on 18 July, said a large part of his meetings last week were to discuss a new common approach to immigration in the UK and EU.

“You cannot deal with this Brexit issue unless you deal with the questions of immigration and the anxieties that gave rise to it,” he said. “Since 2016 and the referendum, immigration is Europe’s problem. Today it is upending the politics of every single country in Europe – every single one. There are lots of things we as the UK can do without displacing the basic principle of free movement.”

Clegg has also tried to explain how the UK could participate in the European parliamentary elections in May, and have MEPs leave later if necessary.

The remain diplomats are studying the commission’s plans for the political declaration on the future relationship between the EU and the UK, seen as critical to whether wavering Tory MPs eventually support Theresa May’s deal. The declaration, due to be endorsed by majority vote by heads of government, will sit alongside the legally binding withdrawal agreement covering issues such as money.

It is assumed that the less detail in the declaration, and the more the difficult decisions are deferred, the easier it will be for the Brexit-supporting MPs to wave the deal through on the basis that the UK will be over the legal finishing line. Once outside the EU, more maximalist negotiating positions can be readopted, with or without May as leader.

Clegg was one of the first to predict, as far back as December 2017 in the FT, that the political declaration may be a fudge, in effect leaving the critical aspects of the future relationship, including terms of UK access to the single market, to be negotiated during the 21-month transition period.

How strictly to set the parameters on the future trading relationship in the political declaration, and the legal enforceability of whatever is agreed, are among the biggest issues exercising European capitals, Clegg judges.

Remainers doing their diplomatic rounds are struck by how fearful some in Europe are of Boris Johnson. “Some are trying to find some vacuity or weak verbiage to keep Theresa May in office, so they will look for a kind of merger between the Chequers plan and Canada free trade agreement – Cheqada,” said Clegg.

He said it would be “a democratic outrage to embark on a blindfold Brexit”. Yet People’s Vote officials know it is a threat. “It is harder to denounce a blancmange,” said one, especially if Brussels is the co-chef.

Barnier largely regards his job as negotiating the exit, including the Irish border issue, but not the future relationship. Some member states are inclined simply to bundle the UK out the door and park the controversies so they can get on with the reform of Europe.

Remain lobbyists are trying to glean whether Angela Merkel will try to save May by settling for what Lord Mandelson says would be “cosmetic certainty” in a vague declaration on future relations.

Macron, it is thought, is most apprised of the need to pin down the future relationship if he is to save his vision of Europe in the European elections. Brexit has to be seen as a warning to the populists, not an enticement. Polls showing the unpopularity of a blind Brexit have been sent to the French.

• This article was amended on 17 September 2018 to change a reference from heads of state to heads of government.