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Sterling edges down but still on track for best month vs dollar since 2010

* Graphic: World FX rates in 2018 http://tmsnrt.rs/2egbfVh

* Graphic: Trade-weighted sterling since Brexit vote http://tmsnrt.rs/2hwV9Hv

LONDON, Jan 31 (Reuters) - Sterling slipped against the dollar and euro on Monday on news that European Commission officials had rejected the City of London's proposal to strike a post-Brexit free trade deal on financial services.

The pound was still heading for its strongest month against the dollar since July 2010. It has climbed around 4.7 percent since the start of the year as the U.S. currency has weakened broadly and as investors have become optimistic about Britain's economy and Brexit deal prospects.

But it slipped on Wednesday after Reuters reported that EU officials had told British financiers they won't agree to a deal that would allow finance companies to operate in each others' markets without barriers, because Britain has said it will leave the single market.

Having earlier been trading slightly higher on the day, sterling slipped to the day's low of $1.4127 after the news, down around 0.2 percent.

"That's definitely knocked the pound but I'm not that surprised by the news - I don't think it's going to be a game changer for sterling," said Nomura currency strategist Jordan Rochester.

"At this stage the UK is yet to even agree to a transitional deal and we haven't yet started talking properly about future arrangements," he added.

Leaked analysis showed on Tuesday that Britain's economy would be worse off after Brexit whether it leaves the EU with a free trade deal, single market access, or with no deal at all. It was another blow to Prime Minister Theresa May.

Against a stronger euro, sterling slipped half a percent to 88.16 pence.

The pound strengthened on Tuesday after Bank of England Governor Mark Carney said the central bank was turning its focus back to the more conventional business of bringing down inflation, as Britain's economy showed it was getting over the damage wrought by the 2007-09 financial crisis.

Carney pointed out that wages were gradually rising, something the central bank wants to see as it considers when to follow up on November's first rate hike in a decade.

"We sensed a more upbeat tone to his comments, which suggests that it might not take a lot for the BoE to alter its forward guidance," wrote Derek Halpenny, MUFG's European head of global markets research.

"The risks are piling up that the BoE will be much more active than implied by current market pricing." (Reporting by Jemima Kelly; Editing by Matthew Mpoke Bigg)