Speaking to the Telegraph newspaper, Michael Taylor, the head of equities at Threadneedle Investments, said that Mr Green was not up to the job of leading one of the world's largest financial institutions.
Commenting on a recent investor meeting with Mr Green, Mr Taylor told the newspaper: "We had Stephen Green in here two weeks ago, and, cor, he was asleep on the job is how I would describe it. He's just not up for it."
Mr Taylor added that HSBC's apparent lack of interest in Asian investments was also a worrying development, with Standard Chartered currently being preferred because of its Asia-focused strategies.
The comments follow a recent poor spell for the HSBC, with the bank's share price having fallen by nearly ten per cent in the last two months.