The newspaper cited a top-10 shareholder who said investors had become disillusioned with BP's choice of successor, Paul Skinner, chairman of Rio Tinto
Other shareholders still back Skinner, the FT said.
A BP spokesman said no decision had been made yet on a replacement for Sutherland. The Chairman said last year he expected to stand down around the time of the company's AGM in April.
(Reporting by Tom Bergin; editing by James Jukwey)