Arsene Wenger has already confided that it would been hard not to bid for the striker should he become available, while Goal revealed on Tuesday that Rooney was open to a move to north London.
|8/1||Wayne Rooney is 8/1 with Coral to join Arsenal
When asked if the Gunners could afford a £20 million deal, plus £200,00-a-week wage, Gazidis told reporters: "Of course we could do that. We could do more than that.
"We have a certain amount which we've held in reserve. We also have new revenue streams coming on board and all of these things mean we can do some things which would excite you.
"If Arsene thinks signing big names is the right thing to do, yes. The fact is this year we are beginning to see something we have been planning for some time, which is the escalation in our financial firepower.
"We can look at some options that weren't really in our financial capability. It's a progression over the next two seasons. It's quite significant for the club."
While Gazidis refused to name any transfer targets - despite Goal revealing the Gunners have triggered Everton midfielder Marouane Fellaini's £22m release clause - he believes Arsenal's ambition can see them emulate Champions League winners Bayern Munich.
"We should be able to compete at a level like a club such as Bayern Munich. I'm not saying we are there by any means, we have a way to go before we can put ourselves on that level," Gazidis continued.
"I think we are an extraordinarily ambitious club.
"This whole journey for the past ten years has been about putting us up with the best in the world and now the question is turning that platform now into on-field success."