During Solomon's CEO tenure, the stock has soared 83%. But a string of recent issues has intensified the spotlight on him. The bank sprang into damage-control mode in recent weeks after a March Wall Street Journal story about Goldman's lack of female leadership, including a stream of women exiting under Solomon. One of those executives, Stephanie Cohen, led the bank's big bet on the consumer space. After losing billions of dollars, Goldman is now retreating from the business.

"Solomon's foray into the consumer realm has been met with missteps and steep losses, which seem to have trickled into further human capital issues," ISS wrote in the report on Wednesday.

After Goldman's governance committee evaluated the dual CEO-chairman role in December, the board reaffirmed that it was "the most effective leadership structure," per the company's annual-meeting materials.

Goldman didn't respond to a request for comment from Business Insider sent outside normal business hours.

ISS recommended voting for Goldman's slate of directors, including the next lead independent director, David Viniar. He joined the board in 2013 after stepping down from the bank, where he was most recently the chief financial officer.

"Some may question the decision to elevate a former Goldman executive to the role at this time," ISS wrote.