Bristol Live says Peterborough look set to have plenty of clubs inquiring about Clarke-Harris' availability, and the Blades look to be among them.
Bournemouth and West Bromwich Albion have also reportedly been taking a look at the striker in recent weeks, while scouts from Rangers were at Peterborough to watch him score his treble at the weekend.
The Blades have also been keeping a close eye on the former Bristol Rovers striker, and are interested in bringing him to Bramall Lane as they prepare for life back in the Championship next season.
United are hoping Clarke-Harris could have a similar impact to former Peterborough striker Ivan Toney, who has proven to be a huge success with 28 goals for Brentford as they chase a long-awaited promotion to the Premier League this season.
The Blades already have a connection to Clarke-Harris, with new Under-23s coach Graham Coughlan having managed him during his time in charge at Bristol Rovers.