European leaders nominated IMF chief Christine Lagarde last week to succeed Mario Draghi as president of the European Central Bank, raising the question of who would in turn replace her at the Washington-based Fund.
Without citing a source, the daily said Berlin and Paris had originally agreed to support Carney with a view to him taking over at the IMF in 2021, but this had been moved forward due to Lagarde's forthcoming move to the ECB.
The newspaper noted that Carney was available from January.
A spokesman for the Bank of England declined to comment on the report.
A French official said at the weekend that, while France was aware that support was growing for Carney, who holds Canadian, British and Irish citizenship, there was concern that appointing "basically a Canadian" would set a precedent.
Carney was born and raised in Canada. He has also served in the past as a governor of the Bank of Canada.
The head of the IMF, whose members include most countries in the world, has always been a European, although in the past, large and emerging economies have challenged that practice.
The United States, despite being the IMF's largest financial backer, does not usually field a candidate because, under an informal deal with European partners, it gets the head of the World Bank - the IMF's sister organisation in the Bretton Woods system forged after World War Two.