Mr Johnson last night suggested it was for Brussels to make concessions, saying there was “a deal to be done” if the EU was ready to move.
Asked if the UK could get a deal in the next 10 days, he said: “I very much hope that we will, but obviously that depends on our friends and partners across the Channel. I think there is a deal to be done, if they want to do it.”
Downing Street played down the prospects of a breakthrough today, describing the phone call with Ms von der Leyen as a chance to “take stock” of the situation.
Face-to-face talks between EU chief negotiator Michel Barnier and his UK counterpart Lord Frost are expected to resume in London next week, with just over 50 days to go before Mr Johnson’s self-imposed deadline to get agreement or crash out in a no-deal Brexit.
Mr Byrne told BBC Radio 4’s Today programme that “big issues” remain to be settled on fisheries, the internal market and a level playing-field on standards.
“I personally don’t expect that there will be major progress today,” he said.
“But at the same time, I think it’s very good that the top two are talking. I think that’s really positive.”
He dismissed suggestions that the talks could herald a move to direct leader-to-leader negotiations between Mr Johnson and key EU players like German chancellor Angela Merkel and French president Emmanuel Macron to get a deal over the line.
“Michel Barnier speaks on behalf of all Europeans, individual member states don’t,” said Mr Byrne. “That’s the way the European Union works, the negotiations will not be between any individual member state or member states, but rather between the European Union and Britain.”
Asked if a Biden victory would create a “new dynamic” in the negotiations, Mr Byrne said: “It’s certainly possible.”
Mr Biden was “very clear” in his response to Mr Johnson’s Internal Market Bill that any trade deal between the UK and US would be “contingent upon respect for the Good Friday Agreement and preventing the return of the hard border”, he said.
The UK took “a serious international hit” from the publication of the Bill, which would allow London to override the legally-binding Northern Ireland protocol signed by Mr Johnson with Brussels last year, said Mr Byrne.
He pointed out that politicians on both the Democrat and Republican side in the US have condemned the bill.
And he said it also represented “a practical problem for a trade agreement” with Brussels, because the European Parliament has said it will not ratify a deal so long as the threat of the Bill remains.
Work needed to be done by the UK “at a very quick pace” between now and the New Year to install the physical infrastructure to implement the new checks agreed by Mr Johnson, he said.
Dublin was not envisaging border checks between North and South, but it was a matter for Britain to ensure that this is achieved, said Mr Byrne, adding: “It is absolutely essential that Britain is ready for every eventuality to comply with legal obligations under the protocol”
“This is a very, very serious issue,” said Mr Byrne. “It’s a key issue that will need to be addressed very, very soon, because we’re coming towards the end of the deadline.”
With or without a deal, the UK’s departure from the EU single market and customs union on 31 December will create “huge difficulties” on both sides, he said.
“It’s not, in my opinion, in the interests of British jobs, it’s certainly not in the interest of Irish or European jobs, and the situation wouldn’t be remedied with a trade agreement,” said Mr Byrne.
“I am assuming that that’s the way the British government will look at this issue. It is very much vitally in Britain’s interest that a trade agreement be reached with the European Union.”