Liverpool hope to sign the Germany international this summer with Werner providing a telling reminder of his fearsome ability in front of goal with a hat-trick in a 5-0 win over Mainz on Sunday.
Chelsea have also been linked with a move for the 24-year-old while United remain in the market for a more long-term option up front having signed Odion Ighalo on loan in January after missing out on Erling Haaland.
"Everywhere, I could see him everywhere. I mean, under Jurgen Klopp he would be perfect there [at Liverpool] especially with his hard working," Hargreaves told BT Sport.
"But I could see him at Manchester United, I think they could do with a number nine and his ability to score goals - with Lukaku leaving [last season] - could be a big plus. I think he could fit into that young dynamic Chelsea team. Whoever gets him is going to be lucky."
Werner's hat-trick on Sunday saw him become Leipzig's all-time top goal scorer and extended his tally for this season to 30 goals in 38 games across all competitions.
"Those are superb numbers and any team that plays attacking football, he's going to be perfect. I just hope he picks the right option because he's going to have a lot of them."