The UK left the EU on 31 January but remains under EU trading rules until a transition period ends on 31 December.
Both leaders were called in after negotiators for the two sides said “significant divergences” remained following a week of intensive talks.
“If there is still a way, we will see,” EU negotiator Michel Barnier said.
Main points include fishing rights, rules on state subsidies for business and arrangements for policing any deal.
One source close to the negotiations on the UK side suggested there had been a more optimistic outlook earlier in the week but pointed to demands for EU fishing boats to have 10-year access to UK waters as one issue that derailed progress
Boris Johnson and European Commission President Ursula Von Der Leyen are to speak later as they try to break the post-Brexit trade deal stalemate.