Lagarde's term is due to end in October 2027 but some fear that the far right might win the French presidential race in the spring of 2027, complicating the selection for the new leader of Europe's most important financial institution.
Citing a person familiar with the matter, the FT said Lagarde had not decided on the exact timing of her departure but was keen on Macron and German Chancellor Friedrich Merz to be the key deciders in who succeeded her.
Macron cannot run again for a third term.
"President Lagarde is totally focused on her mission and has not taken any decision regarding the end of her term," an ECB spokesperson said.
The FT report comes only a week after Bank of France Governor Francois Villeroy de Galhau said he would step down in June, more than a year before the end of his term, allowing Macron to name his replacement before the presidential election that the far-right could win.
While it will be up to all leaders from the 21-nation euro zone to pick Lagarde's successor, past practice suggests that any successful candidate must have both German and French support to clinch the role.
There are no formal candidates for the job yet but several names have been floating among ECB circles as potential ECB presidents.
The most prominent among these are former Dutch central bank chief Klaas Knot and Bank for International Settlements general manager Pablo Hernandez de Cos.
Lagarde's non-renewable term at the ECB runs until October 31, 2027.
Before heading the ECB, she was managing director of the International Monetary Fund from 2011 to 2019 and before that, the French finance minister.
