The decision of the British people to leave the European Union could be reversed next year if France and Germany agree that the UK can take control over immigration while staying in the EU single market, the former Labour cabinet minister Lord Adonis said on Sunday.

With concern over the government’s handling of Brexit growing ahead of a key parliamentary vote on Monday, the peer said Angela Merkel, who is expected to be re-elected as German chancellor later this month, and French president Emmanuel Macron could well make such an offer if they believe it could mean the UK remaining in the EU.

Writing in the Observer, Adonis said he believes a majority of peers in the House of Lords will support an amendment to the EU withdrawal bill – now passing through the Commons – requiring another referendum before Brexit takes effect, with the options being to accept the deal on offer, or stay in the EU.

Such an amendment for another national vote, Adonis said, would stand a good chance of being passed by the House of Commons because Labour would by then have reason to support it, and sufficient pro-EU Tories would also rally behind it, he argues.

“The interplay between a referendum and such a Merkel-Macron ‘offer’ will be vital,” he writes. “If it is clear by next summer that Britain is going to hold a referendum, then the incentive for them to make a bold offer greatly increases.”

He adds: “A lot depends upon whether the alternative is the status quo – or EU membership without freedom of movement in respect of right to work and right to reside for all EU nationals. If Chancellor Merkel and President Macron make an offer, probably over the heads of the British government, for the UK to stay in the economic institutions of the EU but with national control over immigration, then I believe the referendum can be won.

“Why might Macron and Merkel make this offer? Partly because – in Macron’s case – he (rightly) doesn’t believe that unrestricted free movement of labour is integral to the single market. Partly because many other EU leaders agree with him. And partly for the big strategic reason – which weighs on strategic thinkers in Berlin – that, if Britain leaves the EU, 80% of Nato resources will then be outside the EU, which is hardly a recipe for European security and stability if you are looking across at the Russian and Chinese bears.”

While Theresa May is expected to avoid any significant Tory rebellion over the EU withdrawal bill at the second reading stage on Monday, there is growing concern among MPs of all parties at the prime minister’s plan to leave the single market and customs union, and the lack of progress in negotiations with Brussels. On Sunday around 30,000 people marched on Westminster demanding that the UK stays in the EU.

Adonis’s intervention also comes amid signs that opponents of a hard Brexit in all the main parties are ready to work together to amend the bill, both to ensure that the option of staying inside the single market is kept open, and that parliament, at the very least, has a binding vote on the final deal before Brexit happens in March 2019. The Observer understands that meetings about how to thwart a hard Brexit have already taken place between senior Labour figures, the Liberal Democrats and Scottish Nationalist party MPs and pro-EU Tories.

Writing on guardian.com, the SNP’s Brexit spokesman, Stephen Gethins, says the kind of cross-party co-operation that led – through a referendum 20 years ago – to the creation of the Scottish parliament should be repeated now to avoid the economically devastating effects of a cliff-edge hard Brexit. Gethins says “we need to build consensus around key issues in Westminster”, including single-market membership and the rights of EU nationals in the UK, adding: “The government is making a mess of withdrawing from the EU. That affects us all. Hung parliaments require opposition to step up and governments to listen, and the SNP in Westminster will work with others and make our voices heard.”

Meanwhile, 33 Labour MPs have written to the Brexit secretary, David Davis, challenging him over his pledge last year that by now the government would be concluding negotiations on “a large round of global trade deals with all our most favoured trade partners”.

In July 2016, Davis said talks would begin on 9 September 2016, and that “I would expect that the negotiation phase of most of them to be concluded within between 12 and 24 months. So, within two years, before the negotiation with the EU is likely to be complete, and therefore before anything material has changed, we can negotiate a free trade area massively larger than the EU.”

In the letter – coordinated by Labour MP Chuka Umunna and the Open Britain campaign group – the MPs challenge Davis to admit that this pledge was “misleading”, and was symptomatic of a Brexit strategy that has been “characterised by delusional wishful thinking”.