Update 6.52pm: Theresa May's plan to maintain a soft Irish border after Brexit could be changed to accommodate European Union concerns, her deputy said, amid signals that Brussels could reject it.
Cabinet Office minister David Lidington, the British Prime Minister's de facto number two, said her high-profile Friday speech was an "ambitious opening bid" for negotiations on trade.
After Tánaiste Simon Coveney suggested the EU could block her plans to maintain a soft Irish border while leaving the customs union, Mr Lidington accepted it could be changed to accommodate concerns.
But he rejected suggestions that the UK would have to accept the fallback option of keeping Northern Ireland in an effective customs union with the EU.
The issue is holding up agreement on Britain's exit terms and a transition deal.
Mr Lidington backed Mrs May's plan to avoid a hard border through technological solutions and placing no new restrictions on the 80% of cross-frontier trade carried out by smaller businesses, while suggesting it could be tweaked.
Asked if the backstop option was inevitable, Mr Lidington told ITV's Peston on Sunday: "I'm much less pessimistic than you are, clearly we are at the start of a negotiating period and will want to sit down with our EU partners and work through where their concerns, whether legal or technical, are and see how we might together address this."
Earlier: Tánaiste: Theresa May 'hasn't given any more detail than we've already heard' about post-Brexit border
Theresa May's plan to ensure the continuation of a soft border between Northern Ireland and the Republic of Ireland is likely to be rejected by the European Union, Tánaiste Simon Coveney has suggested.
The British Prime Minister has committed to leaving the EU customs union which guarantees tariff-free trade, but insists a hard border can be avoided through technological solutions and placing no new restrictions on the 80% of cross-frontier trade carried out by smaller businesses.
However, Tánaiste Simon Coveney told BBC One's The Andrew Marr Show he was "not sure that the European Union will be able to support" the plan, as it would be worried about protecting the integrity of the single market.
"While of course we will explore and look at all of the proposed British solutions, they are essentially a starting point in negotiations as opposed to an end point," he said.
The Tanaiste says the British Prime Minister has offered no more detail than already heard about how to avoid a hard border after Brexit.
Mr Coveney has welcomed Mrs May's commitment to the Good Friday Agreement and the peace process.
He said: "We certainly welcome the fact that she was definitive in terms of the Good Friday Agreement, which is the foundation stone for the peace process.
"But beyond that, she hasn't really gone into any more detail than we've already heard, in terms of how she is going to solve the problem of maintaining a largely invisible border."
Mrs May, speaking on The Andrew Marr Show, said "quite a few" trade deals have included financial services commitments in them, adding that the "very important role" of the City of London for the UK and EU needs to be recognised.
She said: "If we were to accept passporting, we'd just be a rule-taker - we'd have to abide by the rules that were being set elsewhere, and given the importance of financial stability of ensuring the City of London, we can't just take the same rules without any say in them."
Asked for her message to the financial services companies, Mrs May said: "What we're looking to develop is a relationship that means they can stay here in the UK as part of the City of London, that they will be continuing to provide their services across the European Union but they will know - given the sums of money involved, given the importance of financial stability, given the risk that actually the UK bears as a result of having the City here - that it's important that we do that on the basis of recognised regulatory standards, but we can't just accept rules made elsewhere without us having a say in them."
In the interview, which was recorded after her speech on Friday, Mrs May said Britain would align with EU rules in areas like car manufacturing to maintain access to markets but diverge in areas such as fisheries and agriculture.
She said: "Crucially, Parliament will be able to take decisions about the rules that are set and so in the circumstances in which, say, the EU changed a particular rule, there would be a decision for us to take - did we accept it in the future or not?
"But if we didn't accept it, there'd be an arbitration mechanism, an independent arbitration mechanism, so that people would look at it and say actually, you know what, if the UK doesn't accept that, does it make any difference to the trading relationship? And they might say no it doesn't, so there's no consequence, they might say yes it does, and so there would be a consequence."
Mrs May said she was pleased that Taoiseach Leo Varadkar agreed to sit down alongside the European Commission and UK to look at her proposals to maintain a soft Irish border in more detail.
The PM declined to defend Boris Johnson's comparison of the border to crossing between congestion zones in Camden and Islington, but insisted she was "absolutely clear" that there will not be a hard border.
"We've got proposals as to how we're going to achieve that, now we're going to be able to sit down and talk with others about how we're going to do that," Mrs May said.
Mrs May sidestepped a question over whether a Commons vote on the customs union would amount to a motion of confidence.
An amendment to the Trade Bill calling on the PM to try to negotiate a customs union with the EU, tabled by Conservative former minister Anna Soubry, has attracted the support of enough Tories to threaten defeat for the British Government.
When asked what happens if the Commons tries to tie her hands, Mrs May said: "First of all, we're going to be having a discussion with members in the House of Commons because what I have set out today (March 2) in terms of a future customs arrangement with the EU I think is what most people actually want to see."
Asked if she could stay as PM if she lost the vote, Mrs May replied: "What we're doing in looking at the customs issue as we go through these various Bills in Parliament is what is the right customs arrangement for the United Kingdom to have with the EU in future that ensures we can have tariff-free and as frictionless trade as possible across the border."