Earlier this month, Mr Day, who owns Edinburgh Woollen Mill group, acquired 26 million shares in the women’s fashion retailer at 11.445p a share, bringing his stake to 52.4%.
He was then required to make a mandatory cash offer for the shares he did not own.
But Bonmarche’s board said that the offer “materially undervalues [the business] and its future prospects”.
They added: “In view of Spectre’s position as the majority shareholder in Bonmarche, the board has sought to engage with Philip Day to discuss the future plans for the business for the benefits of all stakeholders.
“The board continues to seek positive engagement with Philip Day and looks forward to discussions in due course.
“The board will be writing to shareholders with its formal response to the mandatory cash offer once the offer document has been posted by Spectre.
“In the meantime, Bonmarche shareholders are strongly advised to take no action in relation to their Bonmarche shares.”
It comes after a torrid fourth quarter’s trading for the cut-price womenswear chain, culminating in a profit warning in March – its third in six months.
Bonmarche, which has more than 300 outlets in the UK, had been forecasting an underlying annual loss of up to £4m, but in March it said it expected to be in the red by between £5m and £6m for the financial year to the end of that month.
Chief executive Helen Connolly had said in December that trading conditions were “significantly worse even than during the recession of 2008-9”.
Sales just after Christmas – helped by heavy discounting – had initially left it in a good position but the group said trading since the beginning of March had been “significantly weaker, reversing sales gains made in the previous months”.
The retailer’s board said on Friday that, in light of this, it had been planning “a number of cost reduction actions across the group and anticipates starting the implementation of these shortly”.