Chief executive Ivan Gazidis said the Gunners could swoop for the England forward and perhaps even bigger names as they enjoy a jump in income from sponsorship deals.
Asked if meeting a prospective transfer fee of some £20 million and £200,000-per-week salary for Rooney was now within the club's budget, he said: "We could do that, we could do more than that.
"We have a certain amount which we've held in reserve and we also have new revenue streams coming on board.
"All of these things mean we can do some things which would excite you. We can think about all kinds of things.
"We are moving into a new phase where, if we make our decisions well, we can compete with any club in the world."
The Deloitte Annual Review of Football Finance 2013 showed the Gunners were behind only United and Chelsea in terms of revenue of £235m for 2011/2012.
That figure is to further increase as the North London club feel the full effect of their move to the 60,000-seater Emirates Stadium, which generates around £3.3m per match.
Arsenal also are understood to be able to tap into additional revenue from new commercial deals which are now coming on board - which could be as much as £70m, moving them up into the same territory as European Champions Bayern Munich.
"We've seen two clubs this year in the Champions League final (Bayern and Borussia Dortmund), both of whom run responsible financial models and they're pretty fantastic teams, very exciting to watch. There's no reason why we can't do that," Gazidis added.
"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 long way to go before we can put ourselves on that level.
"But this whole journey over the past 10 years has been with that goal in mind, which is why I say that this is an extraordinarily ambitious club. This has been about putting us up with the best in the world and now the question is turning that platform into on-field success."
Manager Arsene Wenger has operated under financial restrictions for some years and while the club is in the midst of a seven-year trophy drought, he has been applauded by some for ensuring the Gunners are mainstays in the Champions League.
Now, with the new funds available, Gazidis expects great things from the Frenchman.
"It is going to be the players that Arsene believes in," Gazidis added.
"He is pretty blind to price tags, he looks at what he sees with his eyes and makes judgements based on that, and not on reputations and prices...
"That's why I'm excited - because our capability to spend is going up and our efficiency of spend has always been good."
Real Madrid strikers Gonzalo Higuaín and Karim Benzema and Everton midfield star Marouane Fellaini have also been linked with Arsenal.