According to the Evening Standard, executives at the financial institution, which sponsors the project, are unhappy with its direction.
A bank source told the newspaper: "There is increasing feeling among the bank's personnel that Sam can't continue to neglect American playwrights at the Bridge.
"He did Chekhov and Shakespeare in the first year and As You Like It and The Tempest in the second."
They added: "We understand that Sam's philosophy is to make his films in America and his plays British. Our philosophy is we've given the Bridge Project $3 million."
The Bank of America is two years into its three-year sponsorship deal for the project, which is currently on international tour before its residency at the Old Vic this summer.
