Theresa May insisted she had the "determination to deliver Brexit" as she came under pressure from both wings of the Tory party to change course.
Eurosceptics urged the Prime Minister to drop the proposal for a "customs partnership" which they fear would tie the UK too closely to Brussels.
But pro-EU Tories pushed for Mrs May to abandon her red lines and keep the UK in the single market, claiming she would have cross-party support in the House of Commons for such a move.
Meanwhile Jeremy Corbyn also risked unrest within his party over Labour's position on Brexit ahead of a Lords vote on the issue.
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".
She said the UK was "making good progress towards that goal and we will carry on doing so with resolution in the months ahead".
A decision on the Government's preferred customs option has been postponed after Mrs May's Brexit "war cabinet" failed to reach agreement.
A number of Cabinet ministers spoke out against the Prime Minister's hybrid "customs partnership" model which 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.
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".
But pro-EU Tories, seeking to capitalise on the lack of agreement in the Cabinet and a Lords victory on the customs union, pushed for Mrs May to abandon her Brexit strategy and instead commit to a Norway-style approach within the European Economic Area (EEA) and European Free Trade Association (Efta).
Such a move would be unacceptable to many Brexit supporter because it would leave the UK expected to accept free movement, although its advocates claim Mrs May would have some ability to impose conditions.
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.
"But the other reason I think it's a good idea, is that if you look at the EEA terms of reference, it is clear there is more leeway to impose restrictions on freedom of movement."
A Lords amendment which would require the Government to negotiate continued membership of the EEA could be put to a vote on Tuesday, but Labour peers have reportedly been told to abstain.
Labour's Lord Alli, one of the signatories to the amendment, accused the party leadership of being "paralysed by indecision".
He told The Observer: "This is complete cowardice. There is no point in being in politics to abstain, If you stand in the middle of the road someone is going to knock you over."