Birmingham and their Championship promotion rivals Derby County have both offered around £4million for the Mali international forward, although the Blues' bid, which is still on the table, guaranteed a better final figure.
But, according to Press Association Sport, the 28-year-old is reluctant to leave Selhurst Park and he wants to help recently appointed manager Sam Allardyce keep the club in the Premier League.
Sako's wages are also said to be a big stumbling block for any potential suitor, with the £65,000-a-week he is on more than the Birmingham side are willing to pay.
The West Midlands-based club, under new owners Trillion Trophy Asia, want to maintain a sensible structure and not be saddled with huge wages.
Sako is due to travel to Gabon for the African Nations Cup this month with Mali starting their campaign against Egypt on January 17th.