Terim saw his third stint in charge of his country end in July following an altercation with a restaurant owner, and Allardyce confirmed that he was approached to take over the role.
The 62-year-old, who surprisingly left Crystal Palace during the summer, spent just one match in charge of England during his only previous managerial experience at international level, but he insists that he is not missing the job yet.
"[I'm not missing it] at all at the moment. I think my life is very active in talking about football," Allardyce told talkSPORT after confirming the Turkey snub.
Turkey ultimately replaced Terim with former Shakhtar Donetsk boss Mircea Lucescu.