The newspaper reports Lampard wants to sign Mitchell in a £500,000 deal, and hopes the offer of regular first team football can help to secure his services.
And it is claimed United will consider selling the 21-year-old this summer, after he entered the final year of his contract at Old Trafford.
Mitchell impressed while on loan to Hearts last season, and a return to Scotland is an option as he weighs up his next move.
But the chance to test himself at Championship level with Derby could appeal the Mancunian, who played for the same grass roots team as Manchester United colleagues Marcus Rashford and Cameron Borthwick-Jackson.
Derby have also been linked with a move for Chelsea's Jay Dasilva this summer, as Lampard looks to improve his options on the left of his defence.
The Rams have Craig Forsyth, Marcus Olsson and Ikechi Anya capable of playing in Mitchell's position.