Theresa May has insisted she was right not to "give away" a guarantee of continued residence rights for EU citizens in the UK ahead of the launch of Brexit negotiations.
The concession would have left British expats "high and dry" over their own rights to live, work and receive healthcare in the remaining 27 EU states, said the Prime Minister.
Reports this week claimed that German chancellor Angela Merkel rebuffed Mrs May's plea for an early agreement on reciprocal rights when they met in Berlin on November 18.
And European Council president Donald Tusk rejected a plea from 80 British MPs to intervene to end the deadlock.
In a letter, Mr Tusk said he wanted to "avoid a situation where citizens become 'bargaining chips' in the negotiation" but insisted that the issue can only be addressed once withdrawal talks begin under Article 50 of the EU treaties.
The row came as Bank of England governor Mark Carney called for UK businesses to be told "as much as possible, as early as possible" about the Government's Brexit plans.
The Bank said in a report that it was in the interest of both Britain and Europe's financial stability for the Brexit process to be "orderly".
A senior MEP indicated that the question of reciprocal rights could be dealt with early in the two-year process triggered by Article 50.
Paulo Rangel, the Portuguese vice-president of the European People's Party grouping in the European Parliament - which includes members of Mrs Merkel's CDU party - told the BBC's Daily Politics programme: "I believe some chapters - including probably this one, which is probably not so difficult as the others - can be decided before the final agreement or even before a transitional agreement.
"I don't exclude that we can have these reassurances before difficult negotiations in other chapters but it has to be put in the context of the negotiations."
Mrs May told MPs at Prime Minister's Questions in the House of Commons that Mr Tusk's remarks showed she was right to resist pressure from Labour and some Tory backbenchers to offer the guarantee to EU citizens before securing equivalent guarantees for British expats.
The PM has said she hopes the issue can be examined "at an early stage" of negotiations, due to commence in March.
But aides have repeatedly declined to be more explicit about her intended timetable for agreement.
Mrs May said: "I think it is right that we want to give reassurance to British citizens living in the EU and to EU citizens living here in the UK.
"But I think the reaction that we have seen shows why it was absolutely right for us not to do what the Labour Party wanted us to do, which was to simply give away the guarantee for rights of EU citizens here in the UK, because, as we have seen, that would have left UK citizens in Europe high and dry."
In his letter, Mr Tusk told the British MPs: "Just like you, I would like to avoid a situation where citizens become 'bargaining chips' in the negotiation process.
"In order for this not to happen, we will need precise and comprehensive solutions which, other than nice-sounding expressions, will provide citizens with genuine guarantees of security."
Mr Rangel said that Portugal's president Marcelo Rebelo de Sousa was assured during a recent trip to London - when he spoke to Mrs May and Foreign Secretary Boris Johnson - that "there would be no problem with European citizens, and in particular with Portuguese citizens".
But he told Daily Politics that there could be no discussions before the invocation of Article 50, which Mrs May has said she will deploy before the end of March.
"No-one is gaming or playing about this, this is very clear," said Mr Rangel.
"The UK decided it wanted to leave the EU. They know the rules.
"The UK is the country of the rule of law, and the rule is they have to trigger Article 50, so it's impossible to have negotiations with 27 states without accomplishment of the rules."
Asked when the expat issue can be expected to be settled, Mrs May's official spokeswoman said: "This is an issue that we are very clear matters to Europeans here in the UK and Brits in Europe.
"Therefore, in the spirit of working constructively together and having that kind of mature relationship with our European partners, it's an issue that should be settled early on.
"It's a two-year process once we trigger (Article 50) so these are going to be lengthy negotiations.
"But if you look at where you can provide certainty to people, this is an area where we would like to see early agreement."