The France forward admits he is keen on a move to the Premier League with Newcastle United leading the chase for the player, but Montpellier president Louis Nicollin would prefer to keep his prime asset.
Giroud set up Karim Ait-Fana's late winner against Lille on Sunday and there is apparently a release clause that enables the player to leave for £9.5 million.
With Montpellier needing to win at Auxerre on Sunday to secure the club's first Ligue 1 title, Giroud said: "I don't want to drop a bombshell by saying I want to leave. English clubs make me dream, but I'm not obsessed by England."
But Nicollin was quick to point out. "He'd be very, very expensive.
He added: "The style of play in England would suit him. But he'd be very, very expensive. I hope to keep everybody, even Giroud who is very attracted by England.
"A player is like a singer. If he's badly advised then everything goes wrong. Giroud is a lad surrounded by good people, especially his wife, and he'll go far.
"It will be hard to keep Giroud. But if it's to go and join Newcastle, well I'm not in agreement with that. That doesn't please me. The departure clause in the contract exists without existing but does exist."
Giroud's agent has apparently also held recent talks with Champions League finalists Bayern Munich.