The two leaders – who spoke for over an hour on Saturday in an attempt to break the stalemate – are due to hold a further phone call on Monday evening to assess whether an agreement is possible.
Irish premier Micheal Martin warned that the talks were on a “knife edge”.
“My gut instinct is that it is 50-50 right now. Things are on a knife edge and it is serious,” he told RTE.
“I don’t think one can be overly optimistic about a resolution emerging and my sense, having spoken to some of the key principals here, that this is a very challenging issue to resolve, particularly around the level playing field.”
With time rapidly running out before the Brexit transition period concludes at the end of the month, the chief negotiators Lord Frost and Michel Barnier are meeting in a last-ditch attempt to resolve the remaining issues.
In a joint statement following their call, Mr Johnson and Ms von der Leyen acknowledged “significant differences” remained on fishing rights, competition rules and the mechanisms for resolving disputes.
“Both sides underlined that no agreement is feasible if these issues are not resolved,” they said.
“Whilst recognising the seriousness of these differences, we agreed that a further effort should be undertaken by our negotiating teams to assess whether they can be resolved.”
Ahead of the meeting however, British sources warned there was no guarantee they would succeed.
“This is the final throw of the dice,” said one UK source close to the negotiations.
“There is a fair deal to be done that works for both sides, but this will only happen if the EU is willing to respect the fundamental principles of sovereignty and control.”
Mr Eustice also confirmed that Government is going ahead with plans to bring back to the Commons legislation enabling it to override elements of Mr Johnson’s “divorce” settlement with Brussels in breach of international law.
It means that on Monday – when Mr Johnson and Mrs von der Leyen are due to be speaking – MPs will vote on whether to overturn amendments by the House of Lords which removed the provisions in the UK Internal Market Bill relating to the Irish border.
MPs will then go on later in the week to consider the Taxation (Post-Transition Period) Bill which contains further similar provisions.
The legislation has infuriated the EU and – if it is passed – could further sour the mood in the negotiations making a deal harder to reach.
However Mr Eustice said: “These clauses are very important, particularly in the event that we leave without an agreement.
“It is absolutely crucial that the UK Government has the powers to provide the legal clarity and the legal certainty that business will require at that point.”