Sky News has learnt that a number of leading institutional investors plan to write to the board of Victrex to demand Andrew Dougal's resignation unless he steps down voluntarily in the coming weeks.
On Wednesday afternoon, John Wood Group announced that Richard Howson, the former Carillion chief executive, had stepped down as a non-executive director with immediate effect.
Victrex defended Mr Dougal's position, telling Sky News that it remained supportive of him and insisting that it had not yet been approached by any shareholders seeking his departure.
One major investor, however, said Mr Dougal's role at Victrex was now "untenable".
"The audit chair role is vital.
"There is no way we can have confidence in someone who held that post at Carillion for six years leading to it going bust."
The shareholder said he expected Mr Dougal to resign imminently.
A former chief executive of the building materials group Hanson, Mr Dougal also chaired the audit committee of Taylor Wimpey (LSE: TW.L - news) , the housebuilder, when it was brought to the brink of collapse after the 2008 banking crisis.
He was paid £56,600 for his directorship of Victrex last year.
The auditing of Carillion's accounts is to come under scrutiny from the Financial Reporting Council, which is expected to fast-track a probe into the crisis at the outsourcing giant.
A number of other regulators, including the Financial Conduct Authority and the Insolvency Service, will also be undertaking investigations into directors' conduct.
The focus on Mr Dougal's role came on the same day that Theresa May and Jeremy Corbyn clashed in the Commons over the liquidation of Carillion.
Greg Clark, the Business Secretary, was one of several ministers who held talks with leading bankers to warn them against tightening credit terms for Carillion's sub-contractors - many of which could face financial ruin after its collapse.