Michael Gove and other Tory Brexit campaigners on Sunday called on Theresa May to unilaterally guarantee the rights of 3.2 million EU citizens to remain in the UK, as they back a parliamentary report that brands the government policy as “unacceptable”.

Gove is one of several pro-Brexit Tories on the all-party select committee on exiting the EU who say that May’s approach is causing great “anxiety” and “uncertainty” to people who work hard in the UK, pay their taxes and deserve immediate reassurance about their futures.

The committee says, in a report published on Sunday, that EU citizens should not be used as “bargaining chips” in negotiations over Brexit.

The report piles yet more pressure on May, following a vote in the House of Lords last week in which peers defeated the government by 358 to 256 over the rights of EU citizens. The peers backed an amendment to the Brexit bill urging ministers to ensure within three months of article 50 being triggered that the EU citizens living in the UK be given the right to stay.

While ministers have vowed to reverse the defeat and strike out the amendment when the bill returns to the Commons next week, the report is evidence that concerns expressed by Labour, SNP and Liberal Democrat MPs are widely shared in Tory ranks, even among those who have advanced the case to leave the EU.

As well as Gove, other hardline Leave supporters on the committee who unanimously approved the report include former Tory ministers Dominic Raab and Peter Lilley.

So far May has insisted she will not guarantee the rights of EU citizens until the other 27 member states make similar pledges that the rights of UK nationals living in their countries are also assured.

The attempt to turn their futures into a negotiating weapon has angered many senior Conservatives, including Viscount Hailsham, who told peers last week that guaranteeing the rights of EU citizens was a “matter of principle”, as many EU citizens had made “career changing” and “life changing” decisions to move to the UK which could not easily be reversed.

The committee report states: “The result of the referendum and subsequent debate in the UK and across the EU have created a great deal of anxiety and uncertainty for EU citizens resident in the UK and for UK citizens in the EU.

“EU nationals in the UK did not have a vote in the referendum. They came to the UK legally and have contributed to the UK economically and culturally and enriched UK society. The vast majority have worked hard, paid their taxes, integrated, raised families and put down roots. It is difficult to see what more the UK could have asked of them. The result of the referendum, however, has made them very unsure of their future. Although the government has said it wants EU citizens to be able to remain, the committee notes that this has not offered sufficient reassurance that the rights and status that they have enjoyed will be guaranteed.”

The MPs add: “EU nationals in the UK and UK nationals in the EU are aware that their fate is subject to the negotiations. They do not want to be used as bargaining chips, and the uncertainty they are having to live with is not acceptable. Notwithstanding the assurance given by the home secretary, we recommend that the UK should now make a unilateral decision to safeguard the rights of EU nationals living in the UK.”

Facebook Twitter Pinterest Michael Gove is among those saying government policy on EU workers is ‘unacceptable’. Photograph: Jonathan Brady/PA

The committee also calls for the system by which EU nationals can apply for permanent residence after having lived in the UK for five years to be drastically simplified. EU nationals currently have to fill in an 85-page form that the MPs say is so “complex and onerous” that it will cause huge bureaucratic logjams. The Labour chair of the committee, Hilary Benn MP, said: “The permanent residence application process is disproportionately burdensome and involves collection of information which goes far beyond what is required to prove residence over a five-year period. While there will always be complex cases that require detailed consideration, it should be possible to clarify the status of the vast majority of individuals already here by simply using a streamlined system.”

Labour’s Brexit spokesman, Keir Starmer, said on Saturday: “This just emphasises how isolated the prime minister is now on this. Whether you campaigned to remain or leave, there’s a growing consensus – including now the House of Lords and the cross-party Brexit select committee – that EU nationals should not be used as bargaining chips. The prime minister needs to listen to this, and act now.”

Friends of Gove and Raab said they still expected them to vote against the Lords amendment when it returned to the Commons, on the grounds that the concession should be made separately and should not have to be included in the actual Brexit bill to trigger article 50.

Labour, while insisting that it will not seek to block or hold up Brexit, will continue to support the Lords amendment in the Commons.

On Tuesday the government is facing the prospect of a second defeat on the Brexit bill when peers vote on an amendment that would require a proposed deal on Brexit to be put to a vote in both houses of parliament before it is agreed with the EU, or voted on in the European parliament. The amendment has strong cross-party backing and is expected to be passed by a large majority.