Glynwed shares, at 251p, are at their lowest for more than four years and have underperformed the market by 40 per cent in the last year.
Shareholders also want the Glynwed board to bring in a strong independent outsider to review the decision by Mr Davies, 67, to promote Tony Wilson, another long-serving company employee, to chief executive in a year's time.
Kit Farrow, a non-executive director of Glynwed, would not comment on whether the company had lined up a successor to Mr Davies, while confirming that some leading shareholders are taking a close interest in boardroom changes.
"The institutions have been in touch, though not with me personally," said Mr Farrow, a well-connected executive who is director-general of the London Investment Banking Association. "I will of course be talking to my colleagues."
Ten days ago, Mr Davies, who has been at Glynwed for 30 years, announced a number of boardroom changes but disappointed some shareholders with the slow pace of implementation.
Mr Davies revealed that Bruce Ralph, 57, who Mr Davies helped select, will step down as chief executive in a year. According to analysts, Mr Ralph has never impressed the City since joining Glynwed four years ago from Dowty, when Mr Davies split his combined role of chairman and chief executive.
The Glynwed chairman then announced that although he plans to leave the board, he and his Glynwed colleagues had already chosen Mr Wilson (23 years with the company) to replace Mr Ralph.
"The board is of the view that Mr Wilson is very well qualified," said Mr Farrow.
However, one of the leading institutional shareholders said that the decision over Mr Wilson should have been left to a new incoming chairman.
Shareholders were left in the dark over how soon Mr Davies plans to leave.
He announced at the annual meeting that he would not stand for board re-election again, but he does not come up for a new endorsement by shareholders until 2000.
Shareholders want to know more about what Mr Farrow describes as "self- generated" boardroom changes. "I look forward to talking to the directors," said Max Jourdier, senior investment manager at Britannic Assurance, which owns 4.7 per cent of Glynwed.
Shareholder pressure comes amid discontent over Glynwed's profit performance, slowness in selling under-performing engineering businesses and the sagging share price.
David Blackwood, analyst at Merrill Lynch, recently commented that as a result "sentiment towards the management is poor".
Last year, profits after exceptional charges dropped to pounds 70.1m from pounds 84.2m.
Glynwed's consumer businesses, including Aga and Rayburn, performed strongly but profits slumped at its metals distribution division as stainless steel prices collapsed.Reuse content