Mr Carney took over as Governor in June 2013 for an eight years term, but with an option to leave after five years.
On Saturday The Times reported that he was “more likely than not” to choose to return to Canada in 2018.
It also said that decision could come “within days”.
In response the Bank of England said: “Nothing has changed. The governor has said he will make his decision public by the end of the year.”
Mr Carney has been attacked for his warnings, made before the referendum on European Union membership, that the UK economy would be hit in the event of a vote to leave.
Prime Minister Theresa May has also been critical of the Bank’s stimulus scheme for the UK economy – know as quantitative easing, or QE.
In her speech to the Conservative Party conference, she said that under QE “people with assets had got richer, people without them had suffered”.
Speaking to the BBC on Sunday the Business Secretary said: “I think Mark Carney has done a tremendous job, a fantastic job, during his tenure there. It is clearly a decision for him.”
Mr Clark was financial secretary to the Treasury when Mr Carney was appointed by the then Chancellor the Exchequer, George Osborne, in November 2012.
“I think it was a brilliant appointment,” Mr Clark said.