The British luxury handbag retailer Mulberry has rejected an increased £111m bid from Mike Ashley’s Frasers Group, describing it as untenable.

Frasers, which already owns 37% of the company, increased its offer to 150p a share on 11 October from 130p, which had valued Mulberry at £83m. However, the retailer’s biggest shareholder, Challice, a group controlled by the Singaporean entrepreneur Christina Ong and her husband, was quick to rebuff the proposal, saying it had no interest in selling its shares. Challice’s 56% stake means it can block any deal.

The Mulberry board said on Tuesday that it had considered the proposal carefully with its advisers and decided unanimously that “the possible offer is untenable and that the company should focus its attention on driving the commercial performance of the business”.

It also reiterated its belief that the combination of Mulberry’s new chief executive, Andrea Baldo, a new debt facility and fundraising of nearly £11m would put the company “on a firm footing to ensure we are well set up for future growth”.

The board acknowledged that Frasers had been supportive of the brand through its participation in the fundraising, in which the Sports Direct owner bought £3.9m of new Mulberry shares, and said it looked forward to further interactions with Frasers in the future. The board also noted that Frasers had announced it had sought to engage with Challice directly but said it would not comment.

Mulberry shares were down 4% on Tuesday after the board’s announcement at 123p.

Under UK takeover rules, Frasers must make a firm offer for Mulberry by 5pm on 28 October or walk away.

Frasers, which owns Sports Direct, the House of Fraser department stores, Evans Cycles and the Flannels luxury streetwear chain, pounced on Mulberry after it said it needed to raise cash after a £34m pre-tax loss in the year to the end of March and a slide in sales.

Frasers has said it wants to avoid another Debenhams. It lost £150m on the collapse of the department store chain in which it was a shareholder, and has said it made the bid for Mulberry because it would “not accept another Debenhams situation where a perfectly viable business is run into administration”.

Mulberry, which sells leather goods, hats, scarves and other accessories, has struggled to compete against bigger international brands, especially since the post-Brexit ending of shopping tax breaks for tourists to the UK.

Thierry Andretta, who led Mulberry from 2015, took the brand upmarket, but Baldo, who previously ran the fashion label Ganni, is expected to shift it back to a more broader appeal.

Susannah Streeter, the head of money and markets at Hargreaves Lansdown, said: “Words of frustration have been pumped out from both sides. Frasers Group has clearly lost so much faith in the current direction of the company, it’s fearful of collapse and has warned it does not want to see another Debenhams style scenario play out.

“But Challice is also clearly exasperated by the relentless focus, wanting to give the new CEO, Andrea Baldo a chance to bed in and help revive the brand’s fortunes.

“There are still hopes for a higher bid to come through but whatever the outcome, the bigger price tag slapped on Mulberry’s has enthused investors and hopes for the brand’s turnaround.”