Veysey said the Anfield club are keeping tabs on both players ahead of the summer transfer window:
Oxlade-Chamberlain is a prime target of Manchester City but a source close to the situation has told Football Insider that Liverpool, Chelsea and Manchester United have also asked to be kept informed about his situation.
Manager Jurgen Klopp is also in the market for a new goalkeeper this summer despite sanctioning a lucrative new five-and-a-half-year contract for Simon Mignolet in January.
Szczesny has spent the season on loan at Roma after Petr Cech joined Arsenal from Chelsea last summer to take the No1 spot at Emirates Stadium.
Both players are interesting targets for the Reds. They are naturally gifted, but their talents remain unfulfilled.
Still just 22, Oxlade-Chamberlain is perhaps the biggest disappointment. He's been given every opportunity to shine by Gunners boss Arsene Wenger, but he hasn't been able to shake off the injury bug or develop consistency on the pitch.
From a physical standpoint, the former Southampton winger has everything needed to be a top player. A sturdy build, lightning pace, a fierce shot and excellent technical quality should make Oxlade-Chamberlain an attacker teams fear.
But the problem is there's rarely any end product to go with his raw attributes. He's scored just two goals in all competitions this season while failing to provide even a single assist, per WhoScored.com.
Finding his best position has been a tricky issue for Wenger, who has deployed Oxlade-Chamberlain on either flank as well as through the middle. But it's the lack of game-deciding contributions that's driving the current exit rumours.
John Cross of the Daily Mirror identified the inconsistent attacking midfielder as one of up to eight first-team players who could be sold at the end of the season. The Independent's James Orr also believes Oxlade-Chamberlain is fighting for his future in north London.
Perhaps a move to Merseyside is all Oxlade-Chamberlain needs to finally unlock his immense potential. He certainly has the qualities to fit in the quick-breaking, fluid forward line Liverpool manager Jurgen Klopp favours.
As for Szczesny, he's another infuriating player, one with the ability to be as good as any goalkeeper yet without the temperament to make the most of his talent. He knows few equals as a shot stopper, but the Pole still makes crazy decisions when challenged by crosses into the box.
Really, the 25-year-old has become complacent, rarely showing the willingness to acknowledge faults in his game and improve. He's had run-ins with Wenger over smoking, leading to him being dispatched on loan to Serie A side AS Roma.
But Cross reported for the Daily Mirror in January that Wenger wants Szczesny back to deputise for first choice Petr Cech next season. Yet Klopp's need for more talent between the posts may be greater.
Simon Mignolet's form has been sketchy at best for most of this season. Ex-Liverpool defender Jamie Carragher even told Sky Sports (h/t Ian Doyle of the Liverpool Echo) the Belgian stopper must be replaced in the summer.
Like Oxlade-Chamberlain, Szczesny has the core ability to become a top player. But unleashing that potential would be a tough task for Klopp based on how much Wenger has struggled to do the same.
For £40 million, Liverpool should look for more sure things.