An activist hedge fund that has repeatedly criticized Barclays called on the British bank’s board on Monday to withdraw its support for its chief executive, James E. Staley, over his ties to the financier Jeffrey Epstein.

In a public letter, the investment firm, Sherborne Investors — which says it is now the biggest single shareholder in Barclays — said renominating Mr. Staley as a director was “extremely ill advised.”

The call by Sherborne is the latest point of pressure on Mr. Staley for his yearslong relationship with Mr. Epstein, who killed himself in August after facing new allegations of sex trafficking of underage girls.

Last month, Barclays disclosed that British banking regulators had been examining how Mr. Staley described his business ties to Mr. Epstein, which dated back to Mr. Staley’s role as the head of JPMorgan’s private banking business.