Martin Kohlhaussen, chairman of Commerzbank, and Sir Charles Powell, Lady Thatcher's former foreign affairs adviser, are increasingly concerned for the future of the group. Neither was available for comment, but they are said to be looking inside and outside Tiphook for a chief executive who will at least complement Mr Montague, if he hangs on as chairman.
Mr Montague is believed to have the continuing support of Rupert Hambro, the deputy chairman, and Eric Goodwin, an executive director.
Mr Montague is one of the highest-paid executives in Britain, with a salary last year of pounds 851,000. His position at Tiphook has become increasingly precarious as the company - which has a gearing of more than 400 per cent and is badly hit by recession - runs into deeper financial trouble. It is already in talks with its bankers.
A company source said: 'Some of the non-executives are unhappy with the situation and have been for a while. The resignation of Roger Braidwood (the group's former finance director) was a sacrifice made to appease various interests, but I cannot believe that it has solved anything.'
Tiphook's new public relations adviser confirmed that the possibility of splitting Mr Montague's roles had been on the agenda for a while and that the past week's events might hasten the process. However, one source argued: 'Mr Montague is an all-or-nothing person. He will either hold on to both jobs or leave the group altogether.'
Tiphook's advisers are trying desperately to reassure a sceptical stock market that the company will reach a deal with its bankers after last Thursday's profits warning.
In New York on Friday afternoon, some of the group's pounds 400m of dollar-denominated bonds fell 25 cents in brisk trading to 75 cents as investors complained of a lack of communication from the company.
Tiphook's bankers, led by Commerzbank, Lloyds and NatWest, have a problem on their hands. It is difficult to see how the group can generate enough cash to pay its debts.
The company is still committed to pounds 327m of capital expenditure on new trailers over the next five years, leaving it deeply cash-negative. Abandoning that would lose it the low tax rate it secures from capital allowances.
The dividend, which cost pounds 21.2m this year, must be in doubt, though the high yield resulting from the generous payout has been one of the few props for the company's tottering share price.
There have always been question marks over the value of Tiphook assets, but unloading the fleet of trailers and containers on the market would drive values down further.
The company also has to contend with its bondholders. It has moved a large portion of its debts from bank borrowings into dollar-denominated bonds over the past 12 months. That raises the spectre of legal action from litigious US bondholders.