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Vince Cable has dismissed claims the Tories could be holding back to give their coalition partners a better chance in an upcoming by-election.

The business secretary said the Lib Dems did not have David Cameron's backing for the Oldham East and Saddleworth seat.

But he said the Tory candidate had "no chance".

The prime minister fuelled suspicions of a deal last week when he wished the Lib Dem candidate well.

A Lib Dem victory in the 13 January contest would potentially be a boost to the coalition, easing pressure on the party after its pounding in the opinion polls over student tuition fees.

But any suggestion that the Conservative Party was not fully committed to winning the contest would go down badly with the party's activists and backbenchers.

Asked about Mr Cameron's warm words for the Lib Dem candidate, Mr Cable told ´óÏó´«Ã½ One's Andrew Marr Show: "We have not got his support.

"He is putting up his own candidate and people can compete.

"But I think people in Oldham can do the arithmetic and can see that this is a straight fight between the Liberal Democrats and Labour and they can make their calculations on that basis."

Former Lib Dem leader Charles Kennedy said the by-election would be the first "tangible guide" to how the public viewed the coalition.

But he said it was impossible to call at the moment, telling Sky News: "God knows what's going to happen there."

Lib Dem candidate Elwyn Watkins missed out on the seat by just 103 votes in May's general election, but the result was declared void last month by an election court which found that winning Labour candidate Phil Woolas had lied about him in election literature.

The constituency was a three-way marginal in May, with Conservative Kashif Ali - who is standing again - less than 2,500 votes behind.

Mr Cameron has confirmed he will visit the constituency to campaign for Mr Ali, who he described as "a very strong candidate", and insisted the Tories would be "fighting for all the votes".

But speaking at a press conference at an EU summit in Brussels on Friday, he took the unusual step of saying he wished the Lib Dems well in the contest because of Mr Watkins' experiences during the general election.

Labour has selected Debbie Abrahams as its candidate.

Mr Cable also rejected claims he had fallen out with party leader Nick Clegg over his decision to appear on the Christmas special edition of Strictly Come Dancing.

"He has been very supportive, as have my colleagues," he said.

"I work very hard - sometimes an 80-hour week - I need something to keep me fit and sane. Dancing is my hobby so why not do it in this way?"