Attacking midfielder Adams, 19, joined League One outfit United from non-league side Ilkeston in 2014.
Former Middlesbrough and Nottingham Forest manager McClaren was the Derby boss from September 2013 until the end of the 2014-15 season.
“We spotted one at Ilkeston in Che Adams,” McClaren told The Chronicle. “He ended up at Sheff United. We looked at him when he was at Ilkeston.
“I asked the coaches and they said: ‘He’s not as a good as what we’ve got’, he ends up at Sheff United. And he’s doing alright. You never know, you never know.”
United are eighth in the League One table at the moment with 24 points from 16 matches.
The Blades are just three points behind Bury, who are currently sixth in the standings and occupy the final playoff spot.
Meanwhile, Newcastle are 17th in the Premier League table at the moment with 10 points from 12 games, just two points above the relegation zone.
Derby, now being managed by former Real Madrid first-team coach Paul Clement, are currently fifth in the Championship table with 30 points from 16 matches, just four points behind leaders Hull City.