The Ukip leader has accused Carney of not behaving in an "independent manner" during the run-up to the EU referendum.
In the weeks leading up to the 23 June vote, Carney and the Bank of England issued several economic warnings over what a Brexit could mean for the UK.
Faced with criticism from the Leave campaign, Carney defended the warnings last month, saying the Bank had "a responsibility to explain risks and take steps... to reduce them".
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On Sunday, in an interview with the Globe and Mail in Carney's native Canada, Farage said: "I don’t think the governor of the bank of England behaved in an independent manner during this campaign at all.
"And I think there will be some real questions in parliament about whether it’s appropriate for him to continue in that role.”
A Ukip spokesman told City A.M.: "We believe that we need a new chancellor and we will need a new governor of the Bank of England because the governor has lost his impartiality.
"He ceased to do his job and became part of a political campaign.
"And the governor of the Bank of England should not have done that. It’s like the Queen taking political sides. It is wrong and undermines the impartiality and the seriousness of his position, and he should consider it."
The Bank of England has been asked for comment.