Speaking in Luxembourg, Barnier: "An agreement will be more and more difficult but it is still possible this week."
"Reaching an agreement is still possible but obviously any agreement must work for everyone - for all of the UK and the whole of the EU... It is high time to turn good intentions into legal text," the BBC quoted EU diplomat as saying.
Barnier is scheduled to update EU Ministers on Brexit talks at a meeting in Luxembourg later.
Discussions between EU and UK officials aimed at reaching a Brexit deal have intensified in recent days.
Hopes of a deal being agreed before the October 31 Brexit deadline were boosted after a meeting between UK Prime Minister Boris Johnson and his Irish counterpart Leo Varadkar last week.
The developments come as negotiators stepped up efforts to work out a way to break the deadlock over the Irish backstop, the contingency measure to prevent a hard border on the island of Ireland that is seen as the key factor in the talks.
On Monday, Irish Deputy Prime Minister Simon Coveney raised hopes on an agreement being reached before the EU summit by saying a deal could possibly be achieved as soon as this week.
"But we're not there yet," he added.
With the EU-UK talks in Brussels ongoing, a Cabinet meeting expected to take place on Tuesday has been postponed and is now likely to take place on Wednesday, the BBC reported.
Downing Street sources said it was to allow for a fuller update on Brexit.
The negotiating teams were continuing to try and reach a deal before a crucial summit on Thursday and Friday.
The two-day EU summit is crucial because Johnson must get a new deal approved by MPs by Saturday if he is to avoid asking for a Brexit delay.
According to the Benn Act passed by MPs opposed to no-deal, he must ask for an extension to the Brexit deadline if MPs do not back a deal by then.
However, Johnson has repeatedly ruled out requesting such an extension.
Brexit is due to take place at 11 p.m. on October 31.
( With inputs from IANS )