Speaking ahead of Wednesday's Champions League encounter against United, the Spaniard refused to discuss potentially replacing his former Premier League rival.
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Carrick faces mounting pressure following a six-match winless run, with the Red Devils managing just two victories this season.
However, Fabregas defended the United manager, praising his work and insisting it would be inappropriate to discuss a position currently occupied by someone he respects.
"I won't answer that question because I think that's not a very nice question," the Arsenal legend told the media.
"I don't think that shows enough respect to a great footballer and manager like Michael Carrick at the helm of a wonderful Manchester United side."