* Sterling gains pushes dollar index lower after Brexit comments
* Dollar backs away from 9-month high vs yen
* Euro capped by concerns over Italy's referendum
* Graphic: World FX rates in 2016 http://tmsnrt.rs/2egbfVh (New throughout after moves in sterling)
By Patrick Graham
LONDON, Dec 1 (Reuters) - A perceived crack in Britain's "hard Brexit" line on leaving the European Union dominated trading on major currency markets on Thursday, driving the pound 1 percent higher and helping spur a retreat in broader measures of the dollar's strength.
Sterling surged 1.2 percent to a three-week high of $1.2663, also hitting an almost three-month high of 83.95 pence per euro after Brexit minister David Davis said Britain would consider paying into the EU budget for market access.
As the dust settled, traders were sceptical whether the remarks meant the Conservative administration was getting any closer to giving up the immigration controls that are seen as likely to block such access.
But they were the latest sign of a softer line that offers hope whatever solution emerges in talks starting next year may not be as disruptive to Europe's major economies as previously feared.
"The idea that single market is still a priority is a move away from the 'hard Brexit' line," said Craig Erlam, chief market analyst with retail brokerage Oanda in London.
"Since the two or three major selloffs we saw earlier this year, the Bank of England has changed its stance to holding rates next year, the data has been better and now the rhetoric has softened."
The dollar was roughly steady on the day against the yen after hitting a 9-1/2-month high on the back of a deal to cut OPEC oil output that drove expectations for U.S. inflation and government bond yields higher.
The majority of banks remain focused on the prospect of more gains for the greenback, anticipating that president-elect Donald Trump's mix of tax cuts, spending and trade shifts will raise growth and price pressures in the United States.
But the dollar's failure to push on strongly against the euro - again back above $1.06 EUR= in European trade - hints there may be some fatigue in a rally that dates back two months and has deepened since Trump's victory on Nov. 8.
"You see some brokers on the street with forecasts of parity or below parity (to the euro). For us it is unlikely we will get much more strength," said Constantin Bolz, director for currency strategy with the chief investment office of UBS in Zurich.
"The Fed is likely to hike interest rates next month and the market has fully priced that in, along with roughly two rate hikes next year. Now are we really likely to get three or four?"
The dollar's index against a basket of six major currencies .DXY =USD last stood at 101.23, having risen as high as 101.83 and off a 13-1/2-year peak of 102.05 set last week.
It was almost unchanged at 114.38 yen, down a quarter of a percent on the day from a peak of 114.83 yen. JPY=