Theresa May and Jeremy Corbyn look set to hold further talks over Brexit.
The prime minister struck a conciliatory tone in her response to a letter from Mr Corbyn, which set out his five demands for a Brexit deal.
She said she wanted the two parties to discuss “alternative arrangements” to the Irish backstop – a commitment to avoid a hard border.
But there remains a divide over the customs union, even though she did not reject any of his conditions outright.
No date has been set for the next meeting, but the letter concludes with Mrs May saying she looked forward to the two parties meeting “as soon as possible”.
Meanwhile, Brexit Secretary Steve Barclay and EU negotiator Michel Barnier will hold talks in Strasbourg later, as the EU and UK Brexit negotiating teams discuss proposed changes to the deal.
Writing her response to his letter of last Wednesday, Mrs May told the Labour leader: “It is good to see that we agree that the UK should leave the European Union with a deal and that the urgent task at hand is to find a deal that honours our commitments to the people of Northern Ireland, can command support in Parliament and can be negotiated with the EU – not to seek an election or second referendum.”
This is despite Mr Corbyn repeatedly saying there should be a general election if Mrs May cannot get a deal through Parliament. He has also faced pressure from some of his MPs to push for another public vote on Brexit.
Does the letter give political cover?
By Norman Smith, BBC assistant political editor
It could have been a very different sort of letter.
Mrs May could have just underscored her red lines: No to extending Article 50. No to another referendum. No to a customs union.
Instead, it’s a much more conciliatory and consensual letter.
There’s praise for Mr Corbyn in accepting the priority now should be on reaching a Brexit deal, rather than pressing for a general election.
Praise too for his acknowledgement that the backstop has got to be changed.
And there’s some movement on employment rights and the promise of more cash for hard pressed communities.
Even on the customs union – their key dividing area – Mrs May’s language is more nuanced, even though privately her aides insist there can be no question of accepting a permanent customs union.
It’s unlikely to be anywhere near enough to win over Mr Corbyn.
But it may be enough for those labour MPs in leave supporting constituencies, who are looking for political cover to back or abstain on Mrs May’s deal.
What has Labour asked for – and how has the PM responded?
In his letter, Mr Corbyn asked for five changes to be made to the Brexit deal.
The customs union
The Labour leader called for a “permanent and comprehensive UK-wide customs union” with the EU to be introduced to the deal, with the same external tariff.
He said it would give the UK a say on any future trade deals that the EU may strike.
In her reply, Mrs May said the political declaration – the second part of her deal which is a non-legally binding statement on the future relationship between the UK and EU – “explicitly provides for the benefits of a customs union”, with no tariffs, fees, charges and restrictions.
But, she said, it also allows for the UK to strike up its own trade deals elsewhere.
Mrs May wrote: “I am not clear why you believe it would be preferable to seek a say in future EU trade deals rather than the ability to strike our own deals?”
But, she added her reassurance that securing frictionless trade with the EU was “one of our key negotiating objectives”.
The Single Market
Mr Corbyn also wanted the deal to include a promise for the UK to be closely aligned with the Single Market after it leaves the EU, “underpinned by shared institutions and obligations”.
Mrs May quoted the EU as saying the current deal provides for the closest relationship possible outside the Single Market.
She added: “I am not sure what exactly you mean when you say ‘shared institutions and obligations’, but our teams can explore that.”
The PM also repeated the EU’s warning that completely frictionless trade is only possible if the UK stays in the Single Market.
“This would mean accepting free movement, which Labour’s 2017 General Election manifesto made clear you do not support,” she added.
Labour has called for the UK to stay in step with the EU on rights and protections for workers, which was included in Mr Corbyn’s letter.
On this point, Mrs May said the government had already made commitments on workers’ rights, adding: “We are examining opportunities to provide further financial support to communities that feel left behind.”
This could be referring to proposals that were said to have been discussed earlier this month from a group of Labour MPs in predominantly Leave-supporting constituencies, to allocate more funds to their communities for big infrastructure projects.
The PM also said that while she had “always been clear that Brexit should not be at the expense of workers’ rights or environmental protections”, she did not support automatically following EU rules in these areas.
“Given their importance, we believe these decisions should be taken in our Parliament by our elected representatives,” she added.
But she tried to reassure Mr Corbyn that the government had made legally-binding commitments not to move back in these areas and they are prepared to consider legislating to give these commitments force in UK law.
The Labour leader called for a promise to participate in EU agencies and funding programmes on the environment, education and industry regulation after Brexit.
The prime minister said the government supports participation in EU programmes in a number of areas, as set out in the political declaration – which includes areas such as science and innovation, youth, culture and education, and overseas development.
Mrs May also said the government wanted the closest possible relationship with EU agencies in “heavily regulated sectors”, such as medicines and aviation, “subject to us being outside the single market”.
Finally, Mr Corbyn demanded agreements with the EU on security, such as access to the European Arrest Warrant database.
Mrs May said the government “shares your ambition in relation to security arrangements”.
She said the political declaration secured agreement on the exchange of Passenger Name Record, DNA, fingerprint and vehicle registration data and on arrangements “akin to the European Arrest Warrant to surrender suspected and convicted persons efficiently and expeditiously.”
But, she added, there is a challenge that as a third country outside of the EU, there are restrictions on the UK’s ability to participate in some EU tools and measures.
Mrs May said: “The government does not believe it would be in the interests of either the UK or the EU and its Member States to see a reduction in joint security capabilities.
“Labour’s support for this position going into the next phase will, I hope, send a powerful signal that the EU should reconsider its stance.”
Labour is yet to respond to the letter.