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UPDATE 2-Sterling hits one-month high vs euro on "Brexit breakthrough" hopes

* Graphic: sterling and gilt yields http://bit.ly/2dgAXn1

* Graphic: World FX rates in 2017 http://tmsnrt.rs/2egbfVh

* Graphic: Trade-weighted sterling since Brexit vote http://tmsnrt.rs/2hwV9Hv (Recasts after comments from European Parliament's Brok)

LONDON, Dec 4 (Reuters) - Sterling jumped to a one-month high against the euro on Monday after a member of the European Parliament's Brexit group said there was a "very good chance" of a deal on an initial divorce package between Britain and the European Union.

Elmar Brok said he was "astonished" at how far the negotiations had come and that differences remained over "just a few words".

British Prime Minister Theresa May will have lunch with European Commission President Jean-Claude Juncker and European Union Brexit negotiator Michel Barnier on Monday in Brussels, where she will try to persuade them to start discussions on a new trade pact and a two-year transitional deal.

The European Parliament's Brexit coordinator Guy Verhofstadt sounded a more cautious tone than Brok, telling reporters that the chances of a deal when stood at 50-50, with Parliament still pressing for more from London on guarantees for the rights of EU citizens in Britain.

But traders were optimistic, with sterling reaching as high as $1.3525 after the comments, up from around $1.3440 beforehand.

"To me, this is a Brexit breakthrough and it should be solid for the pound," said Neil Jones, Mizuho's head of hedge fund currency sales in London.

"I sense that we're moving towards a sign-off that will enable the UK to move into trade talks and that's only going to be a good thing for the pound."

Against the euro, the pound climbed 0.8 percent on the day to a 87.68 pence, up from 88.17 pence.

Earlier, better-than-expected economic data showing a modest rebound in British construction activity last month had little perceptible impact on the pound, with the Brussels talks stealing the limelight in London trading.

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 remain fraught, diplomats say.

"If a green light is provided today for talks to move on to future relations, including a timely transition arrangement, it would open the door for further pound gains in the near-term," said MUFG currency economist Lee Hardman.

A Reuters poll last week found economists believe the chance of a "disorderly Brexit" - where no deal has been reached when the two years of talks are scheduled to close in March 2019 - has declined over the past month.

Such optimism drove sterling to six-month highs on a trade-weighted basis last week. It traded around 0.4 percent below those highs on Monday.

Speculators bought back into the pound in the week up to last Tuesday, data showed on Friday, with positioning moving into positive territory - reflecting more bets that sterling will strengthen than that it will fall - for the first time since October. (Reporting by Jemima Kelly; Editing by Peter Graff)