The private equity firm is through to the second round of the auction, alongside rivals Merchant Equity and Argyll.
It is understood that MFI would have to shell out over£200 million to a bidder to snap up the retail chain, which has a pension deficit of£150 million.
MFI revealed two months ago that it had been approached by several parties interested in its retail division. At the group's AGM in May, it revealed orders for the year were down 19 per cent and same-store sales had fallen 8 per cent.
Apax has a 30 per cent stake in Focus.