* May meets Juncker, Barnier over lunch
* Wants leaders to sign off on start of trade talks
* Tusk cancels trip to prepare trade talks (Updates with reports of Irish border deal, Tusk)
BRUSSELS, Dec 4 (Reuters) - British negotiators were reported to have agreed outline rules with Ireland for their border after Brexit on Monday, sending the pound higher on hopes of a deal on EU free trade as Prime Minister Theresa May arrived for crunch talks in Brussels.
Britain's Daily Telegraph and Financial Times reported that London had agreed with Dublin, which has the backing of the European Union, to maintain regulatory "alignment" with the EU on both sides of the border to avoid a possible flare-up in the violence which troubled Northern Ireland for decades.
European Commission President Jean-Claude Juncker spoke to Irish Prime Minister Leo Varadkar by phone. EU lawmaker Philippe Lamberts, who met Juncker earlier on Monday, said the draft text on Ireland would commit Britain to "full alignment" on rules.
Donald Tusk, the EU summit chair, tweeted ebulliently after speaking to Irish Prime Minister Leo Varadkar that there was progress on the Irish issue to unblock UK-EU trade talks: "Tell me why I like Mondays!" the former Polish premier wrote.
"Encouraged after my phone call with (Varadkar)," he added. "Getting closer to sufficient progress at December (EU summit)."
This is code for Britain complying with a set of conditions the EU wants met on divorce terms before leaders will agree to launch talks on a future trade deal at a summit next week.
May and Juncker made no comment to reporters when they met at the EU executive's Berlaymont headquarters for a lunch that diplomats and officials hope can seal a breakthrough that would open the way to negotiations on future trade relations.
Tusk, cancelled a trip to Jerusalem and Ramallah and hastily scheduled his own meeting with May for after lunch. Officials said he was preparing to call round EU leaders to get agreement on trade negotiations. National envoys handling Brexit were also summoned urgently to a meeting in Brussels later on Monday.
London has broadly agreed to many of the EU's divorce terms, including paying out something like 50 billion euros. But the issues of the rights of expatriate citizens and the UK-EU border on the island of Ireland defied a deal until the last minute.
Brussels officials and diplomats sound moderately confident. But they cautioned that much will depend on the outcome of May's talks as to whether the EU will agree on "sufficient progress".
Elmar Brok, another member of the European Parliament who met Juncker and his Brexit negotiator, Michel Barnier, shortly before they met May, said "just a few words" had separated the sides and that there was a "very good chance" of agreement.
In London, May's spokesman said: "Progress is being made. There's more still to agree upon."
Irish state broadcaster RTE said that a draft text would commit Britain to "continued regulatory alignment" with the EU to avoid divergence on the two sides of the Irish border.
Varadkar said he would speak in public shortly.
Dublin, backed by the rest of the EU, is seeking strong assurances that London will commit to keeping business regulations in Northern Ireland the same as in the EU, to avoid a "hard border" that could disrupt peace on the island.
Irish Foreign Minister Simon Coveney said early on Monday that the talks were in a "sensitive place", with the British and Irish governments discussing possible texts of an agreement.
Britain is seeking to keep its options open, having rejected a commitment to leave Northern Ireland in a customs union with the EU or to keep the whole United Kingdom in one.
May depends in parliament on a pro-British party in Northern Ireland that rejects any deal which would divide the province from the British mainland. Ireland and the EU say maintaining a customs union is the best way to avoid "regulatory divergence".
Juncker and Barnier met the European Parliament's Brexit team to brief them on progress. The legislature, which must approve any withdrawal treaty if a disruptive Brexit is to be avoided in March 2019, has demanded that EU courts have the final say in guaranteeing rights for 3 million EU citizens in Britain. Britain insists that it will no longer accept the supervision of the European Court of Justice. (Additional reporting by Andrew MacAskill, Estelle Shirbon and Kate Holton in London, Alastair Macdonald, Elizabeth Piper and Jan Strupczewski in Brussels and Conor Humphries in Dublin, writing by Alastair Macdonald; Editing by Larry King, Peter Graff and Andrew Heavens)