
The Saddlers have been negotiating with O'Donnell since November and are understood to have offered him a two-year deal that would see him become one of the high earners at Banks's Stadium.
City manager Phil Parkinson is in the market for another stopper to compete with Ben Williams but will not discuss possible targets.
Championship side Reading and soon-to-be League 1 outfit Millwall are also believed to be monitoring the situation, with several other sides also on the look-out.