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LONDON, Jan 9 (Reuters) - Sterling was on track for its biggest daily losses in three months on Monday, sliding around 1 percent to a 10-week low to the dollar after Prime Minister Theresa May said she was not interested in Britain keeping “bits” of its EU membership.

In her first televised interview of the year on Sunday, May denied Britain would face a “binary choice” between curbing immigration and having preferential access to the bloc’s single market when it leaves the European Union.

May re-asserted that line on Monday, saying it was wrong for investors to interpret her comments as pointing to a “hard Brexit” - in which border controls are prioritised over market access - when asked about sterling’s falls on the back of her weekend interview.

The prime minister added that she rejected the idea of a “hard” versus a “soft” Brexit, that the government was seeking “the best possible deal” in terms of access to the single market, and that she had said nothing new on Sunday.

Investors, though, took no comfort.

By 1650, sterling was trading at $1.2166, down 1 percent on the day and close to a 10-week low of $1.2125 touched earlier. That put it on track for its worst day since Oct. 11.

“The market just takes the most negative view - it assumes that because May is keeping her cards as close as possible to her chest, there is no plan for Brexit and that we’re going to ‘hard Brexit’ out of the EU,” said BMO Capital Markets currency strategist Stephen Gallo.

Another risk was also playing into sterling’s weakness, strategists said - that of Scotland calling another referendum on independence.

“Scotland’s First Minister Sturgeon said that she is not ‘bluffing’ regarding the prospect of another independence referendum if Scotland is ‘driven off a hard Brexit cliff’, something that may have also fueled the sterling tumble,” said IronFX currency strategist Charalambos Pissouros.

Against the euro, sterling tumbled 1.3 percent to an eight-week low of 86.95 pence.

“May saying that it’s not about keeping ‘bits’ of the EU suggests it’s not going to be about keeping access to the single market,” said HSBC currency strategist Dominic Bunning.

“She said we will have full control of our borders, and given what the other side of the debate - the EU - has said, that’s not compatible with full access to the single market, free movement of capital, free movement of goods and services. That’s the direct trade-off that the FX market is looking at.”

May also used Sunday’s interview to dismiss criticism from Britain’s former EU ambassador Ivan Rogers that her government’s thinking on Brexit was “muddled”, but she ignored growing calls from business leaders, lawmakers and the parliamentary opposition to provide more details of her exit strategy.

Sterling has fallen 19 percent against the dollar since Britons voted to leave the EU on June 23 and around 12 percent against the euro, which has itself been vulnerable to political uncertainty. (Editing by Gareth Jones and John Stonestreet)