Attempts have been made to sell the business over the past six months and the decision to close was made after no buyer came forward, said Mr Woollons.
Last week Woolworths' administrators Deloitte said there was still interest in parts of the business but added it had "not come close" to finding a buyer for the firm.
British Airways took over BMI from Lufthansa earlier this year and has since grounded a number of BMI Baby routes as it prepares to close that part of the business in September after failing to find a buyer.
Since both the buyer and seller are represented by agents in most transactions, brokers must collaborate to close deals at the same time as they compete for listings.