Keir Starmer, the Labour leader, is coming under pressure from Europhile MPs and party activists to support sweeping changes to the Brexit deal as concern rises about the damage it is doing to Britain’s economy and jobs and the freedom to move and work across the continent.

A report for the leftwing group Another Europe is Possible and separate research by the non-aligned, internationalist Best for Britain organisation both strongly support the case for more active engagement with the EU to improve the deal and rebuild relations with member states.

Starmer, a Europhile and leading proponent of a second referendum under Jeremy Corbyn’s leadership, has been criticised by some in the party for accepting Boris Johnson’s Brexit too willingly, and refusing to speak out more strongly about its damaging effects, since it passed through parliament with Labour backing last December. Asked at the time if Labour could push to renegotiate the deal at the next general election, Starmer said he did not want to still be debating Brexit. “If we are still arguing in 2024 about what has gone in these past four years, we’re facing the wrong way as far as I’m concerned,” he said.

But with the deal damaging British exporters as well as contributing to tensions in Northern Ireland and restricting people’s right to travel and work in the EU, many pro-Europeans are looking for Labour to lead a push for a better deal short of rejoining the bloc.

In a report published by Another Europe is Possible, Luke Cooper of the London School of Economics’ foreign policy thinktank LSE Ideas, argues that the trade deal badly needs revisiting to protect jobs, workers’ rights and the economy and restore membership of EU projects such as the Erasmus student exchange programme. One result of the Brexit deal, he says, is that Britain is having to replicate agencies and other bodies it used to be part of when an EU member. Trade and economic cooperation with Europe would also be far easier if Britain agreed to harmonise more regulations with Brussels instead of misleading the public into thinking that the UK was benefiting economically from winning back sovereignty.

University students would like the Erasmus exchange programme to be restored.

His report argues that the UK should actively re-engage. “This should include developing a close working relationship for education and training, rejoining the Erasmus and Erasmus+ programmes and remaining a member of [the research programme] Horizon.”

A UK governmentBritain should also seek a mutual rights agreement with the aim of “restoring the rights of UK citizens to work and study in the EU as part of a reciprocal arrangement”.

Nadia Whittome, Labour MP for Nottingham East, said: “This important new report shows what a disastrous trade deal the Conservatives have struck with the EU, and why we shouldn’t back away from campaigning for an alternative. We need a deal that is committed to high regulation, protects manufacturing jobs, and restores citizens’ rights and freedom of movement.”

With pro-Remain parties tempering their criticisms of Brexit and the Brexit deal for fear of alienating anti-EU voters, the former campaign group Best for Britain is relaunching this week to champion the cause of internationalism, including closer cooperation with the EU.

A poll conducted for the group has revealed that many voters, including those in Labour Leave constituencies, are broadly supportive of greater international cooperation in several areas.

The research shows that 59% of voters in Labour Leave seats say that now Britain has left the EU, we should cooperate in areas of mutual benefit. Only 5% say we should avoid cooperating altogether. Naomi Smith, chief executive of Best for Britain, said: “There is an urgent need to improve the current deal with the EU, so it delivers for the country as a whole. Our latest polling shows that all voter types, including those in majority Leave-voting seats, want the UK working with European allies to defend our shared interests. The clearest path to power for Labour is to look outwards and adopt strong, internationalist positions, that this data clearly proves will resonate with the electorate at large.”