The forward, whose Fratton Park contract expires next week, is yet to sign the Blues contract that has been presented to him – despite previous pleas from fans to do so.
He remains the only out-of-contract player yet to commit his future to the club, with Sean Raggett, Michael Jacobs and Reeco Hackett penning fresh deals during the close season.
Cowley has previously said the presentable offer on the table fits firmly into Pompey’s budget for the season ahead.
However, he told The News this week that it won’t be there indefinitely.
Meanwhile, chief executive Andy Cullen has claimed preparations are in place in O’Brien decides to turn his back on the Blues.
That puts the ball back in the Irishman’s court, with the former Sunderland man not present at Roko after the players returned for pre-season training.
Yet, sections of the Fratton faithful believe if a decision is not made very soon, the club should say goodbye and wish the player well for the future.
Others, though, are taking a more hard-line approach and think his time is already up.
Here’s how fans on social media platforms Twitter and Facebook have reacted to the ongoing impasse...
Adam Newell: As much as I would love Aiden to sign if it’s not done my Monday move on.
Michael Cox: Well I would like him to re sign, but if money is his soul motivation then it’s best we let him go elsewhere, sooner rather than later.
Paul Collins: Move on, go and get a player for PUP.
Steve Metherell: If he leaves us dangling then bin him off, if all he’s driven by is money and better contracts we’re not the right team, if he wants support and atmosphere we are! Deadline looming for him surely then let’s move on.
Stephen Morby: We need to move on. I'd be surprised if he signs. He's obviously not happy with the deal on offer.
Steve Mac: Can O’Brien actually perform back to back 90 mins?
Already got Thompson & Jacobs that struggle badly with that issue - can’t afford to many luxuries if we are to again go with a small squad.
@Imposter66: He's not interested in the project as seen by his reluctance to sign. He's had his chance...move on.
@RobTroubadour: If he doesn’t want to take the contract then move on to the next player, don’t wait it out.
I don’t see much movement elsewhere for him.
@CharlieFarnsba9: Off you pop Aiden, if you wanted to be here you’d have signed by now. Bye.
@jeffharris_665: Got until Monday eve for me.
@Willmott3Sam: Unpopular opinion but I’m not too fussed about O’Brien coming back.
Don’t get me wrong he would be a great experienced squad player but not first team starter.