The 30-year-old left-back, who was released by Blues at the end of last month, played for the Blades in their 1-1 pre-season friendly draw at Rotherham on Friday night.
Clapham recently had a trial at Midlands neighbours Wolves and it is believed that they have offered the former Ipswich Town player a contract.
Should he land a deal with Neil Warnock's side, Clapham would team up with another former Blues player in Ian Bennett. The former Blues goalkeeper moved from Leeds United to Sheffield United earlier this week.