Theresa May and cabinet ministers have expressed “strong disappointment” at a string of fresh defeats in the House of Lords over the EU withdrawal bill, with Liam Fox accusing unelected peers of trying to block the UK from leaving the European Union.

Ministers said they were concerned that the amendments risked tying the government’s hands behind its back in negotiations with Brussels at discussions during its weekly meeting, Downing Street said.

The prime minister and the Brexit secretary, David Davis, told the cabinet the government would be “robust” when the bill returned to the House of Commons, her spokesman said, saying it was “vital to ensure that the legislation is able to deliver the smooth Brexit which is in the interests of everybody in the UK”.

Speaking earlier on Tuesday, Fox said it would be rash for Labour MPs who represent leave-voting constituencies to back the Lords’ amendments to the EU withdrawal bill when it returns to the Commons.

The government suffered three more Brexit defeats in the House of Lords overnight, taking the total to nine, and Fox’s appeal to Labour MPs suggests the government is banking on the support of opposition Brexiters to quash the changes.

The key defeat was an amendment that would allow parliament to send ministers back to the negotiating table if MPs voted down Theresa May’s Brexit deal.

Labour said the new clause to the EU withdrawal bill would in effect prevent Britain crashing out of the EU with no deal and would give Tory remainers the confidence to vote against a damaging deal. The cross-party amendment, supported by 19 Conservative rebels, succeeded with a majority of 91.

Fox said on Tuesday: “I think there’s a very big debate now about whether the unelected house can actually thwart the view of the British electorate in a referendum and … legislation coming from the House of Commons.”

He conceded the parliamentary arithmetic was difficult in a hung parliament.

“We don’t have a parliamentary majority and that makes life harder, but there are of course quite a number of Labour MPs who represent seats where there is a very heavy vote in favour of leaving … who I think are rash, to say the least, if they try to confront the democratic view of the British people,” he told BBC Radio 4’s Today programme.

Downing Street would not commit to saying the prime minister was confident she would overturn all of the defeats in the Lords. “We will look at the amendments, but the fundamental point is that the bill left the House of Commons in a way that could deliver a smooth Brexit and that is how we want the bill to complete its course,” the spokesman said.

“What is important is that it keeps the government’s hands open on negotiations with Brussels.”

The slew of Lords defeats, and the difficulties they will pose in the Commons, will cast a shadow over a crucial meeting of the Brexit cabinet subcommittee this week, where senior ministers are split over the customs union.

Key Brexiters want to drop one of the two options put forward by the government. Fox and Boris Johnson have called on May to abandon her preferred option of a “customs partnership”, which relies on Britain collecting EU import tariffs on behalf of Brussels.

The option would bring the UK closer to solving issues over the Northern Irish border but Brexiter critics have said it would be a customs union in all but name.

Fox said on Tuesday: “I don’t think there is a customs union that could ever be acceptable. If we are in a customs union of any sort we will have less ability to shape Britain’s future than we have today. That is not what the public voted for.”

Fox refused to deny he would resign if a customs partnership was agreed, and did not challenge the assertion by the Today presenter John Humphrys that May was a “horribly weakened prime minister”, replying that a hung parliament had made agreeing its Brexit legislation harder.

Overnight, peers also voted for an amendment that would require parliamentary approval for the Brexit negotiating mandate in phase two of the talks – similar to the process where EU27 leaders approve the mandate of their chief negotiator, Michel Barnier.

The third defeat was led by the Labour peer Alf Dubs, who won backing for his amendment that would ensure child asylum seekers would be allowed to join family members in the UK after Brexit.

A Liberal Democrat attempt to amend the bill to give voters a referendum on the final deal did not pass, although one Labour frontbench peer, Philip Hunt, ignored party orders to abstain and voted in favour. Labour sources said he would be hauled before Labour’s leader in the Lords and chief whip, but the Conservatives called for him to be sacked.