David O'Leary's men are preparing for Sunday's derby encounter with Birmingham City, but the Turkish defender was given permission to stay away from the training session.
The 28-year-old angered many English observers with his actions during the Euro 2004 qualifier on Saturday, as he twice goaded David Beckham.
After already alienating some Villa supporters by making a mocking gesture to them when scoring in the match with Charlton Athletic, the defender has publicly admitted his future with the Midlands outfit is uncertain.
O'Leary could make a diplomatic decision over Alpay's exclusion from the starting line-up at the weekend as Ronny Johnsen is back to full fitness and could slot in at centre back.
The former Fenerbahce stopper was given Monday off to recover from the exertions of Istanbul, but was expected to join his team-mates on Tuesday.
Alpay failed to attend, which led to fresh speculation about his future, but the club have confirmed he was given permission to sit-out the session.
With him missing two days of training already this week, doubt has been cast over whether he will even be included in the squad for the noon St Andrews showdown.
