Rose takes the role after investing a small amount of his own money in Blue Inc last year, following a larger investment from the billionaire Reuben brothers.
Since leaving the top role at M&S last January, Rose has also taken an advisory role at private equity group Bridgepoint and invested in online firm The Hut.
He is also a non-executive of South African food and clothing retailer Woolworths.
Blue Inc’s chief executive Steve Cohen’s private vehicle Marlow Retail bought the menswear chain in 2006 from the Levy Family, who founded the business in 1912.
The chain traded as Mr Byrite until 1999, when it was rebranded as Blue Inc to attract a younger audience.
The business, which has 191 stores and employs 2,000 staff, is on track for sales excluding VAT of £80m and earnings of £5.5m this year.