The former Gunners defender says injuries have prevented the club's Three Lions stars from fulfilling their potential over the last couple of years.
The Gunners had just two players in Roy Hodgson's squad for the friendlies against Germany and the Netherlands at the end of last month, in the form of striker Danny Welbeck and out-of-form forward Theo Walcott.
Walcott has been a regular target of criticism from Arsenal fans this season for a perceived drop in performance level, and his compatriots such as Alex Oxlade-Chamberlain and Kieran Gibbs have struggled with injury and poor form.
Meanwhile, Jack Wilshere a notoriously injury-prone player but a star at club and international level when fit, ha snot played a single minute of football since scoring twice for England against Slovenia last June.
Keown believes the fact that those players have not been able to all get regular game time for the club together on a consistent basis this season has held them back, and insists they must prove that they can overcome their injury problems.
The former centre-back said in his weekly 'Tackle Keown' Q&A for the Daily Mail: "We've hardly seen them all on the field together in the past couple of seasons, and I think that's down to injuries more than anything.
"Wilshere and Walcott's injury histories are well documented, and Oxlade-Chamberlain and Gibbs have suffered their fair share as well. [Carl] Jenkinson seemed to be doing well at West Ham, but know he's picked up cruciate ligament damage.
"One of the main things you have to do as a player is prove to your manager that you are reliable in terms of your fitness and injury record. Unfortunately, this promising British core haven't really been able to do that yet and, subsequently, haven't lived up to their true potential."
Walcott appears likely to be in Hodgson's squad for Euro 2016 regardless of his relatively underwhelming campaign, whilst Welbeck's four goals in 10 club appearances since returning from his own long-term injury have seen him force his way back into the England manager's plans.
Meanwhile, there has been considerable debate in recent weeks about whether Wilshere should be taken to France for this summer's tournament even if he returns to action in the next few weeks after such a long absence.
Hodgson's comments at the start of last month that the midfielder can still earn a spot in his squad sparked something of a backlash from some quarters, with former Gunners midfielder Ray Parlour amongst those who insisted the head coach must be extremely careful when deciding whether or not to select the 24-year-old.