Basturk was a member of the Bayer Leverkusen side that reached the 2002 Champions League final but his recent career has stalled.
Allardyce told the club's official website: "This is about trying to extract a top-quality player's ability, resurrect it.
"He has not been playing regularly for some time and been cast aside by his club in Germany. He wanted to come and prove himself in the Premier League and for us to have the opportunity to give him that platform is quite exciting.
"We will put him through his paces in the next two or three weeks to bring him up to speed physically and then hopefully with desire and determination he could be a major player for us at the back end of the season."