Pressure is mounting on Sir John Peace to relinquish at least one of the three FTSE 100 chairmanships he holds.
The 64-year-old City stalwart is chairman at Asia-focused bank Standard Chartered, luxury retailer Burberry and credit-checking group Experian.
City institutions say Sir John "must no longer spread himself thin" across the three companies following an embarrassing gaffe at Standard Chartered last week. This saw him make a grovelling apology for remarks he made earlier this month which suggested the bank had not deliberately broken United States sanctions.
On Thursday, he was forced to say: "I made the certain statements that I very much regret and that were at best inaccurate.
"Standard Chartered bank unequivocally acknowledges and accepts responsibility, on behalf of the bank and its employees, for past knowing and wilful criminal conduct in violating US economic sanctions laws and regulations, and related New York criminal laws, as set out in the deferred prosecution agreement."
Leading investors in the banking group say they have raised the issue with the company over the past few months and plan to do so again in future meetings. "It's dangerous to juggle multiple roles in an economic climate like this," one told The Independent. "Imagine the perfect storm of issues all blowing up at once. How could anyone dedicate enough time to each business?"
Another said: "I think most investors now agree that chairmen should really only chair one FTSE 100 company at a time.
"Sir John has been highly impressive at Standard Chartered and it is hard to criticise him. However, it's also hard to dispute the argument that bank chairmen, in particular, should not focus on more than one role. Especially in the current regulatory environment."
Temasek, Standard Chartered's largest shareholder, is not believed to have raised the issue with the company, although it is understood to have discussed separate concerns about the board's size, which it believes is too large.
A spokesman for Standard Chartered said: "Sir John is committed to doing at least three days a week with us and sees it as his primary role."
Experian said it had recently conducted a search for a new chairman and decided Sir John was the best man to lead the company, while Burberry declined to comment.
Last year Standard Chartered was fined $667m (£441m) by the US Department of Justice and New York Attorney General for major breaches of sanctions against Iran and three other countries.
Standard Chartered declined to comment any further on its dealings with US regulators although sources last week said Sir John's apology had been issued following extensive discussions with the Department of Justice.
"They tried to play hardball with the US regulators and lost," said Simon Maughan, analyst at Olivetree Securities.
Sir John was paid $1.82m by Standard Chartered in 2011 and £374,000 by Experian and £350,000 by Burberry last year.