The discussions, intense and secret, are the last attempt to conclude a new trade partnership, for the moment when the transition of the exit of Great Britain from the EU will end at the end of this year.
If the two sides overcome differences of opinion, the new agreement will cover all aspects, from trade and energy to transport and fishing. If they fail, trade in goods and services, valued at $ 900 billion a year, will be affected from January 1 by tariffs and quotas.
A European diplomat and a British official said negotiations would continue face to face in Brussels on Monday, after a weekend of intense talks.
An update on the progress and chances of reaching an agreement is expected on Wednesday or Thursday, the sources added.
Another European diplomat said over the weekend that talks were still difficult on most difficult topics.
However, both sides have previously signaled their readiness for a compromise in the fisheries sector, which is a sensitive issue for both the UK and France, as well as a few other EU Member States.