British Prime Minister Boris Johnson and the head of the EU's executive, Ursula von der Leyen, agreed in a phone call on Saturday to step up Brexit talks to close "significant gaps" barring a new trade partnership.
The two sides have said this week's round of talks aimed at getting a new, post-Brexit trade agreement from 2021 made some progress but not a breakthrough.
The EU says a deal must be sealed by the end of the month to leave enough time for ratification by the end of this year.
Johnson and von der Leyen discussed the next steps in their call on Saturday.
"They agreed on the importance of finding an agreement, if at all possible, as a strong basis for a strategic EU-UK relationship in future," they said in a joint statement.
"Progress had been made in recent weeks but ... significant gaps remained, notably but not only in the areas of fisheries, the level playing field, and governance," it added.
The two leaders instructed their Brexit negotiators, Michel Barnier and David Frost, "to work intensively in order to try to bridge those gaps".
More trade talks are due in London next week and in Brussels the following week before the 27 national EU leaders meet on Oct. 15-16 to assess progress.
An estimated trillion euros worth of annual trade would be at stake if they fail to get an agreement.
The EU says it will not implement any new deal if London undermines their earlier Brexit divorce treaty with its draft Internal Market Bill.