The two main shareholders, who between them account for 7 per cent of WH Smith's equity, are said to prefer a candidate with more of a retailing background than Mr Hammill's.
They are also thought to be upset by reports that Mr Hammill is touting a break-up of the group, and that the succession issue has been allowed to fester over the summer without a clear announcement from the board.
"Keith is an excellent finance director, but you do need a retailer running this company at the moment," said a source at one of the two opposing shareholders.
"In any case, if the board are so clearly set on Keith, why has it taken so long for them to come to a decision, and why hasn't an announcement been made?"
The shareholder claimed to have had "no formal meetings" with Mr Hammill on the subject of a demerger. At least one board member is believed to support the two institutions' view, and to be concerned at the way the succession issue has been handled; the board member is pressing for a retailer - perhaps from outside the group - to be appointed instead of Mr Hammill.
There are still three outside candidates with retail experience in the running for the job. Two have not been identified, but one is known to be Stuart Rose, currently a senior executive at the Burton Group.
The internal candidates include Alan Giles, managing director of Waterstones, Richard Handover, currently running the newspaper distribution arm, and John Hancock, managing director of the US retailing operations. All have retailing backgrounds.
A third institution, Franklin Resources, the California-based fund manager which is part of Templeton Worldwide and holds 11 per cent of the equity, is believed to be supporting a break-up bid, following the argument that it is the only way to realise shareholder value. This would involve reopening discussions with Virgin over the sale of Smith's 75 per cent stake in Virgin/Our Price, for which Virgin reportedly offered pounds 135m in May - an offer that was immediately rebuffed by Bill Cockburn, then chief executive of WH Smith.
Analysts were surprised at the size of the offer, which was some 30 per cent above their valuations for the stake.
Under the break-up plan, the company would also explore a possible demerger or trade sale of Waterstones, which could fetch pounds 250m. "The best way to realise shareholder value is to break the group up and retain just the core retail business," said Matthew McEachran, retail analyst at Henderson Crosthwaite. "Then the management can focus on getting the merchandise right and expanding the hotel and airport concessions."
Mr McEachran estimates that a break-up would be worth pounds 1.25bn, or 441p a share. On Friday, the company's shares closed at 378p.
WH Smith sources said this weekend that no strategic decisions would be made until a new chief executive was appointed, and that this decision would not be made until the first week of September at the earliest.
"The biggest priority facing this group is to appoint a new chief executive. Until then, it is fair to say that no strategic decision will be made," said a spokesman.
The company is understood to have earmarked the US music retailing business, which is only just breaking even in dire trading conditions, as a candidate for disposal.