Backbench Labour MPs are urging the party’s leadership to place membership of the single market back on the table in terms of Brexit policy, claiming the general election result has “dramatically changed” the context of the talks.

There is disquiet over the continuing refusal of Keir Starmer, the shadow Brexit secretary, to consider the possibility of full membership of the internal market. Some Labour MPs believe having only access to the single market – rather than membership – would be economically damaging.

Starmer repeated his argument that Labour’s focus should be on the benefits of the final deal, rather than its structure, during a recent interview with the Guardian and on the BBC’s Andrew Marr show.

“We want to retain the benefits of the single market and the customs union. Formal membership, full membership is only available to EU member states and that’s why there’s all the discussion about what sort of model that gets us to close to membership,” he said.

However, some MPs want Labour’s position to contrast more clearly with the government’s.

Stephen Doughty, the MP for Cardiff South and Penarth, said: “The political and parliamentary context for Brexit has changed dramatically. We cannot carry on business as usual. Labour should be out front putting forward a positive vision that keeps all options open and responds to the growing disquiet across the country of [Theresa] May’s reckless Brexit.”

He said there was a growing sense within the parliamentary party and among the Labour membership “that we need to keep our options open on issues like single market and customs union”. Mishandling that could affect the prospects of millions of young people entering the job market over the coming years, he argued.

Ben Bradshaw said he had always favoured continued membership of the single market and customs union, and had made that clear to his constituents in Exeter. He said the election result made this position more appealing for the party.

“The latest polls show overwhelming support for this as well as even more overwhelming support among people who voted Labour,” he said. “I am still hopeful that the Labour frontbench will recognise this new reality and show the vision and flexibility that I think the British public now want.”

Chuka Umunna, the backbencher and chair of Vote Leave Watch, an organisation that seeks to hold politicians to account for promises made during the Brexit referendum campaign, has also spoken out on the issue.

Writing in the Guardian, Umunna argued that Labour’s policy to seek “access to the single market” did not go far enough because it would not secure non-tariff benefits, such as safeguards for workers, consumers and the environment. He argued that while free movement was a central pillar of the single market, it was possible to have a fairer immigration system under the current rules.

“We could move towards ‘fair movement’ as a full member of the EU, [for example] requiring people who have been [here] for three months but haven’t got work to leave, but no government has done that yet,” he said.

“All that Labour people would ask is that we do not ape Theresa May in these negotiations but oppose her when she is unnecessarily playing Russian roulette with people’s jobs by taking the option of single market membership off the negotiating table before having any discussion about it.”

The issue has caused tension at the top of the party. The shadow trade secretary, Barry Gardiner, suggested Labour would push for a bespoke trade deal or “reformed membership of the single market”.

However, he made clear he felt the party’s frontbench was unified, echoing Starmer’s assertion that the key issue was not the structure of the deal but what benefits the UK would draw from it.

Gardiner agreed that the election had changed things – but for the Tory party rather than Labour.

“They can now, in effect, pull the negotiations away from the hard deregulatory agenda that it was on track for and get a much softer form of Brexit,” he said. “But the softer form of Brexit the Conservative party wants is solely about market access and shouldn’t be confused with the softer form of Brexit Labour wants.”

Gardiner said Labour was determined to maintain the social and environmental protections and regulations put in place by the EU. “The key thing is that we get the right outcomes, that we preserve jobs and our economy. We must then embed the social and environmental standards, protections and rights because the way the Tories propose to translate them into UK law would make them vulnerable to watering down at the whim of the secretary of state.”