The 26-year-old was reportedly close to sealing a switch to the Championship club, having been spotted with manager Harry Redknapp, but surprisingly pulled the plugs on the deal at the last minute.
Instead, he signed for Bursaspor on a season-long loan from Italian Serie A side Udinese.
“I don’t think I came to Turkey because of money,” Badu said, as reported by Joy Sports.
“I want to play football and I will get it in a competitive Turkish league.
“The offer I got for Birmingham was very great, it was very difficult to reject it but I needed to taste the Turkish league.
“I spoke with Jerry Akaminko, John Boye, [Emmanuel] Adebayor and Asamoah Gyan before our [pre-season] friendly with Kayserispor.
“There were great reviews and I want such challenges. It is not about money."
Birmingham were so hit by Badu's decision that Redknapp was left lamenting.
"I did a deal the other day for a lad from Udinese - Badu - a Ghanaian boy and top midfield player," the Englishman told his club's official website.
"I drove to Huddersfield to watch him play.
“It took me six-and-a-half hours to get to Huddersfield, met him after and then spoke to him two days ago and he gave me his word on the phone that he wanted to play for me.
"I wake up the next morning and he has signed for a Turkish club!”
Badu was also once on the books of Spanish outfit Recreativo Huelva and local sides Asante Kotoko and Berekum Arsenal.