Coventry Airport has said it would consider offers to take it over. Birmingham International Airport (BIA) has reportedly rejected the chance to buy the lease for an estimated �30m.
Coventry Airport said even if it is sold, low-cost flights from Thomsonfly, which is owned by the airport's operators TUI, would remain.
The leader of Coventry City Council, Ken Taylor, who is also on the board of BIA, said it would not make business sense for Birmingham to buy the lease.
Amended package
He said: "If you're going to that level, you need to invest an awful lot of money and you're taking money from what you could possibly spend on Birmingham."
A spokesman for BIA said the airport company would not comment on speculative reports.
TUI has the operating lease to use the entire Coventry Airport site, including the airfield and buildings, while the city council owns it.
The managing director at the airport, Bill Savage, has denied it has been actively looking for a buyer.
He added: "(In terms of) staying at Coventry in the form of Thomsonfly, there's no question of pulling out. TUI has a long and successful future at Coventry Airport."
The site started running low-cost passenger flights in March 2004.
Last month Warwick District Council gave a conditional approval to plans for a new passenger terminal at the airport.
It agreed to an amended package of measures, including compensation for residents.
A public inquiry, due to start in Leamington Spa in January, will decide whether the development goes ahead.