Theresa May has sought to reassure Brexiteers by restating her commitment to leaving the European Union's single market and customs union after Brexit.
The prime minister told the Sun on Sunday she had "absolute determination to make a success of Brexit".
Mrs May is under pressure from both wings of her party to change course.
The government's decision on its preferred customs option was postponed this week after senior ministers failed to reach agreement over a new model.
One of the government's preferred options - a "customs partnership" - has faced heavy criticism from Conservative Brexiteers.
The hybrid model would see Britain collect tariffs on behalf of the EU for goods destined for the block, with firms potentially claiming back a rebate if products remained in the UK on a lower-tariff regime.
Mrs May has asked officials to draw up "revised proposals" after she was challenged over this plan by cabinet ministers in Wednesday's meeting of the Brexit sub-committee.
The former Conservative Party leader, Iain Duncan Smith, is among leading Brexiteers to again warn Mrs May against a "messy" customs deal.
The Sunday Telegraph quoted a cabinet source saying it would be "unimaginable for the prime minister to press on with the hybrid model after it has been torn apart by members of her own Brexit committee".
The local election results, which saw the Conservatives perform well in many Leave-supporting areas, saw Boris Johnson claim that promising to leave the EU's single market and customs union had been a "key part of Tory electoral success".
However, some pro-EU Tories are still pushing for much closer economic ties and commit to remaining inside the European Economic Area (EEA) and the European Free Trade Association (EFTA) - which would deliver full single market access.
Former Conservative minister Stephen Hammond told the Independent: "It allows you to have some conversations over new regulation, in that you are consulted and are part of the process before it comes in, though it's still not co-determination, of course."
Writing in the Sun on Sunday, Mrs May said she had an "absolute determination to make a success of Brexit, by leaving the single market and customs union and building a new relationship with EU partners that takes back control of our borders, our laws and our money".
"We are making good progress towards that goal and we will carry on doing so with resolution in the months ahead."
Meanwhile, a Lords amendment which would require the government to negotiate continued membership of the EEA could be put to a vote on Tuesday.
However, Labour peers have reportedly been told to abstain.
Labour's Lord Alli, one of the signatories to the amendment, told the Observer that the party leadership was "paralysed by indecision".
What is a customs union?
All EU members are part of the customs union, within which there are no internal tariffs (taxes) on goods transported between them. There is also a common tariff agreed on goods entering from outside.
The UK government has said it is leaving the EU customs union so that it can strike its own trade deals around the world, something it cannot do as a member.
This means the UK and the EU will have to agree a new arrangement for what happens at their border post-Brexit.
The UK, which put forward two alternative proposals last year, has yet to confirm its favoured model.
It is under pressure to make progress on the issue before next month's EU summit.
The government's two options
■? A 'highly streamlined' customs arrangement - This would minimise customs checks rather than getting rid of them altogether, by using new technologies and things like trusted trader schemes, which could allow companies to pay duties in bulk every few months rather than every time their goods cross a border
■? A customs partnership - This would remove the need for new customs checks at the border. The UK would collect tariffs set by the EU customs union on goods coming into the UK on behalf of the EU. If those goods didn't leave the UK and UK tariffs on them were lower, companies could then claim back the difference.
On the eve of the Brexit cabinet meeting, Brexiteers urged Mrs May to abandon the partnership option, presenting a 30-page dossier claiming it would make meaningful trade deals "impossible" to forge and render the UK's International Trade Department "obsolete".
Brexit Secretary David Davis told MPs on Thursday that both options had merits and both had drawbacks "which is why we are taking more time over them".
