A senior ally of Jeremy Corbyn has said holding a second EU referendum could badly affect the relationship between politicians and the public, as a series of Labour backbenchers tabled new amendments seeking such an option.

Ian Lavery, a Labour MP who is also the party’s national campaigns coordinator, said that while the possibility of a second referendum should be maintained as a way to end an otherwise intractable Brexit impasse, he was worried about the consequences.

Labour’s frontbench has tabled an amendment to Theresa May’s Brexit plan which includes this option, as decreed by party policy, Lavery says in an article for the Guardian. “However, we should be in no doubt that asking the voters to vote again on an issue they have already given an answer, until they come up with the right answer, risks serious damage to the relationship between many citizens and politicians at Westminster.”

The best solution would be a Labour government, Lavery wrote, “not rerunning a divisive campaign that seems likely to deliver the same result again and do nothing to answer the demand of a country crying out for real change.”

Such sentiments from someone seen as close to Corbyn’s team will dishearten those in the party keen to push forward the option of a second referendum, something that polls show is backed by a majority of Labour members.

On Thursday a planned cross-party amendment for a so-called people’s vote was dropped by its organisers, who said the lack of explicit Labour frontbench backing meant it was unlikely to pass.

However, the Commons order paper for Tuesday’s crucial series of votes on the next stages for Brexit now features three new amendments tabled by Labour MPs seeking this eventuality.

The amendments, tabled by Ian Murray, Angela Smith and Mike Gapes, are amendments to the Labour frontbench amendment, and seek to mandate the government to organise a second referendum. The Smith version is backed by four other Labour MPs.

In his article, Lavery said May had “spectacularly failed to deliver a Brexit deal that will meet the needs of our country”. He wrote: “Her attempts to revive her categorically rejected deal seem to have about the same chance of success as the kiss of life would have on the T rex exhibit at the Natural History Museum.”

In parallel with Labour MPs’ manoeuvrings over a second referendum, Conservative backbenchers have been tabling amendments seeking to demonstrate their dislike of the Irish backstop, one of the main factors that led to the defeat of May’s plan.

One of the key amendments on Tuesday, if selected for a vote by the Speaker, could be one tabled by the Tory MP Andrew Murrison that has attracted the support of key figures such as Graham Brady, the chair of the backbench 1922 Committee, and May’s former de facto deputy Damian Green.

This calls for the backstop – an indefinite insurance policy that would come into force if no permanent deal can be made to prevent a hard Irish border – to be “replaced with alternative arrangements to avoid a hard border”.

While this could win over many Tory MPs who voted against the deal, and possibly also May’s DUP partners, it would require a change in the EU’s stance, which is seen as very unlikely.

Asked whether more concessions from Brussels were possible, May’s spokesman indicated on Friday that some movement was needed.

“I think there is a clear message from EU leaders that they want us to leave with a deal, and they understand that a deal is in the best interests of the United Kingdom and the European Union,” he said.

“The prime minister put the deal as it stands before parliament, and that deal suffered a significant defeat. So clearly if we are to leave with a deal, we’re going to need to make some changes to it.”