The offer is at 11.445p per share. After the trading update, the shares fell 26%, to a price just below that level.
The Bonmarché board had been holding out against the offer, saying its own cost reduction programme would improve performance.
It said again that the offer “does not adequately reflect the potential longer-term value of the business”, but acknowledged its “poor” trading had changed the picture.
“The increase in uncertainty that has developed, reflecting the trading and financial position of the business during the first quarter of the financial year, makes the certainty represented by the offer potentially more attractive in the short term.”
The terms of the offer are “now fair and reasonable”, the company said.
It said the medium and long-term prospects for the Bonmarché business were good, but it had been told by its auditors that unless trading improved by the time of its results on 26 July, they might include a caveat in its results about its ability to operate as “a going concern” in the long term.
If the deal goes ahead, it could reunite Bonmarché with Peacocks, which was bought out of administration by Mr Day’s Edinburgh Woollen Mill in 2012. At the time, Bonmarché was kept out of the administration and bought by a private equity firm before being floated on the stock market.