* British PM May to meet Juncker, Barnier over lunch
* Monday "deadline" for Brexit deal to unlock trade talks
* EU lawmakers demanded UK let EU court oversee rights
* Discussions "still fluid" with 24 hours till crunch (Adds Weber quotes)
By Alastair Macdonald and Jan Strupczewski
BRUSSELS, Dec 3 (Reuters) - The European Commission added to mounting confidence in Brussels that it is set for a Brexit deal with London when it scheduled talks with EU lawmakers ahead of a crunch meeting on Monday with Theresa May.
However, EU officials and diplomats cautioned on Sunday that it was still unclear that a deal would be struck with the British prime minister when she meets the EU executive.
Two hours before they sit down for lunch with May in Brussels, Commission President Jean-Claude Juncker and his Brexit negotiator Michel Barnier will brief Guy Verhofstadt and his European Parliament Brexit team, an official said.
Verhofstadt and his colleagues wrote to the EU negotiators last week to sound an alarm at what they said were "stalled" talks on EU demands that the rights of EU citizens in Britain be guaranteed directly by the European Court of Justice after Britain leaves the European Union. They also voiced concern about Northern Ireland.
Parliament must ratify any treaty on Britain's withdrawal from the European Union before Brexit in March 2019 in order to create the smooth transition period May wants, and so it will be vital to both sides to keep the legislators on board.
Senior EU officials and diplomats said work was continuing on Sunday. One person close to the discussions said the situation was "delicate". "It's still quite fluid," said a second person involved. "Nothing is agreed until everything is agreed."
Verhofstadt, a former Belgian prime minister and strong critic of Brexit, declined to comment.
The German leader of the centre-right group in the EU parliament warned that lawmakers were not yet satisfied.
"On Brexit negotiations, money is one of the problems, but it is not the biggest one," Manfred Weber said in a statement. "We are much more concerned about the fact that so far negotiations are stalled on the protection of EU citizens' rights after Brexit and on the Irish case," he said.
The EU wants outline accords on three critical divorce terms before it will open negotiations on the transition and a future free trade pact that would follow. May's lunch on Monday is a deadline for the EU to have her final offers before EU leaders consider whether to agree at a Dec. 15 summit to launch Phase 2.
Britain and the EU aim to sign a joint declaration setting out progress toward final deals which the Commission, as the EU executive, would say was "sufficient" for opening trade talks.