Neither the Government nor Aer Lingus has appointed corporate finance advisers to manage the sale of the State-owned airline.
The Minister for Transport, Mr Brennan, said on RTÉ radio yesterday that the Government was looking very closely at the company's options.
But the absence of special advisers indicates that a formal sale process has not yet begun.
It is understood that a change of ownership is not taking up any management time at Aer Lingus, which was prepared for a flotation two years ago before the process was cancelled due to trading difficulties and a stock market slump.
But with Aer Lingus projecting a €45 million profit this year and some sources suggesting that a €50 million profit might be possible before the year ends, it is thought that the Government will look in earnest at the company's prospects in the new year.
Aer Lingus has introduced an aggressive new fare-cutting strategy, meaning it is effectively going head to head with its "no frills" rival Ryanair.
When Mr Brennan met the Aer Lingus board about 10 days ago, he is understood to have said that the Government wanted to assess the company's full-year performance before deciding how to proceed.
Thanks to the airline's recovery, the matter will be addressed much sooner than anticipated. It is no secret that the airline and the Government favour privatisation, although poorly stock markets mean a trade sale is the most likely means of achieving that.
Mr Brennan said on radio: "The question really is what's the best for the future of Aer Lingus? Is it better to keep it as a State-owned company with all that implies or is it better to let it, pardon the pun, fly off into the private sector and take on all its competitors?"
He added: "A year or two back, the Government did decide that it would examine a public offering, that's floating the company. We had to pull it back because market conditions were such in the airline industry that we had to pull it back. Carving out a future for it outside the State sector is probably the preferred way."
Price would not be the only criteria, he said. Despite the crisis in the public finances, this essentially confirms the view of many observers that the Government is less interested in maximising the return to the Exchequer than ridding itself of the airline, which is an almost constant political headache.
Corporate financiers do not doubt that the Government is keen to move ahead with a sale. This in part reflects concern to avail of a window of opportunity in advance of any other downturn in the business.
While a Government decision late last year that selling its entire 85.1 per cent shareholding still stands, unions which halted flights earlier this year are pushing for the end of a pay freeze.