The bid, thought to be in the region of £2.2million, has ended weeks of speculation linking Savage with a move to either Everton or Rovers in the January transfer window.
It remains to be seen whether Everton will counter-offer Rovers, although national media stories indicate Birmingham would want Rovers striker Paul Dickov as part of any deal.
Dickov is still based in the midlands after playing for Leicester City for the past two seasons. However, Dickov is thought to be more than happy at the club and Hughes is unwilling to lose a player who has undoubtedly been one of the stars of the season so far.