The speculation comes ahead of Cordiant's results, which are due in March, and follows the bitter loss earlier this week of its £212m Hyundai US media-planning account. The South Korean car maker consolidated its entire £280m media budget into Aegis Group-owned Carat.
The Hyundai account was recently described by Bungey as one of the global rocks upon which the business is built.
Cordiant today denied speculation that Bungey was under pressure to quit ahead of the results due in March.
Cordiant was among the top 10 fallers on the London Stock Exchange this morning, as it dropped 15.6% to 76p, down from its opening of 94.5p. Cordiant is moving closer to its 52-week low of 49.9p.
Cordiant has refused to comment on the story and told Reuters simply that Bungey has a job to do and "he's getting on with it". Some analysts have poured cold water on the speculation, despite the bad news that Cordiant's results are sure to bring.
According to reports, analysts are expecting Cordiant's figures to include hefty write-downs, principally for the US marketing services group, Lighthouse, which Cordiant was seen to massively overpay for just ahead of the advertising downturn.
The loss of the media business came as part of a double blow for Cordiant, as Hyundai announced it would also be reviewing its creative account, held by Bates USA West, Irvine, California.
Bates has been invited to take part in the creative review, in which Hyundai is being assisted by the consulting firm Select Resources International. It is not clear if the agency will take part in the review.
There has been speculation for weeks now that Cordiant would lose its prized Hyundai US media account. Towards the end of last year, talk that the account was on the move sent Cordiant shares tumbling down, at one point at the end of November by as much as 8.1%.
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