Michael Gove has refused to confirm Theresa May will bring forward her withdrawal bill as planned in the first week of June after MPs from across the political spectrum rejected her 10-point “new deal”.

The environment secretary, who supports May’s deal, said it was a time to “reflect” and consider the options, raising the prospect that the prime minister would have to pull the vote.

In a last-ditch effort to win support, the prime minister offered parliamentary votes on a second referendum and temporary customs union but Labour said it would not vote for the bill and at least 25 Conservative MPs also withdrew their support. The DUP also remains opposed.

Q&A What are Theresa May's 10 commitments in her 'new Brexit deal'? Show 1 The government will seek to conclude alternative arrangements to replace the backstop by December 2020. 2 Should the backstop come into force, the government will ensure that Great Britain will stay aligned with Northern Ireland. 3 The negotiating objectives and final treaties for our future relationship with the EU will have to be approved by MPs. 4 A new workers’ rights bill that guarantees workers’ rights will be no less favourable than in the EU. 5 No change in the level of environmental protection. 6 The UK will seek as close to frictionless trade in goods with the EU as possible while outside the single market and ending free movement. 7 We will keep up to date with EU rules for goods and agri-food products that are relevant to checks at border protecting the thousands of jobs that depend on just-in-time supply chains. 8 The government will bring forward a customs compromise for MPs to decide on to break the deadlock. 9 There will be a vote for MPs on whether the deal should be subject to a referendum. 10 There will be a legal duty to secure changes to the political declaration to reflect this new deal.

Asked whether the bill would be brought forward in early June, Gove told BBC Radio 4’s Today programme: “We will reflect over the course of the next few days on how people look at the proposition that is being put forward … Rather than saying anything precipitate everyone should take an opportunity to reflect and take a look at the bill that is being published later today.”

Pressed on the prime minister’s future, he said: “I think the most important thing we should do is reflect on all the options in front of us … it is always right there is a period of reflection and analysis.”

Gove said May had his “respect and support” but could not say how long she would remain prime minister.

May will defend her proposal in the Commons on Wednesday where the reaction is likely to be very hostile from her own side.

Nigel Evans, a member of the 1922 Committee executive, said the prime minister should recognise it was “game over” and he would press once again for rule changes to allow another vote of no confidence.

Jacob Rees-Mogg, the leader of the Eurosceptic European Research Group of Tory MPs, said he hoped the prime minister would look at the lack of support for her bill and “recognise that in truth the prime minister does not command the confidence of the House of Commons”.

May had sought to woo remainers by promising that if MPs supported the deal at its second reading in early June, she would offer them a vote on a referendum as the legislation passes through parliament – and be bound by the result.

“I do not believe this is a route that we should take, because I think we should be implementing the result of the first referendum, not asking the British people to vote in a second one,’ she said. “But I recognise the genuine and sincere strength of feeling across the house on this important issue.”

MPs would also be given the opportunity to vote on alternative customs proposals, including the compromise offer May made to Labour during the ill-fated six-week negotiations of a temporary customs union on goods. If Labour won the next general election, it could upgrade it to a permanent arrangement.

May made a series of other pledges, including a separate bill to ensure workers’ rights do not fall behind those in the EU, a guarantee that there will be “no change in the level of environmental protection when we leave the EU”, and a promise that parliament will be consulted on the next stage of Brexit negotiations.

But MPs across the House of Commons were not persuaded by May’s new proposals. By Tuesday evening, not a single MP who opposed the deal last time had come out to support it.

The Labour leader, Jeremy Corbyn, said: “We won’t back a repackaged version of the same old deal – and it’s clear that this weak and disintegrating government is unable deliver on its own commitments.”