The pair began working with the first-team again earlier this week following long-term absences with serious injuries.
However, United will face stiff competition for the 27-year-old's signature as the newspaper says Chelsea are also interested in signing Rose when the transfer window re-opens this winter.
First-choice left-back Rose damaged medial ligaments in his left knee during the first half of a goalless Premier League draw against former club Sunderland in January and later underwent exploratory surgery that ruled him out for the remainder of the 2016-17 campaign and beyond.
That has hurt Spurs in that department but Welshman Ben Davies has come into the side and provided great quality, especially this season.
The left-back scored and impressed in the club's last game against Huddersfield.
Lamela has been missing for even longer, having picked up a hip problem that has prevented him from playing since October 25 past year.
Argentina midfielder Lamela, 25, has not played in nearly 12 months because of hip problems.
- Tottenham Hotspur (@SpursOfficial) October 11, 2017Rose on the move?
It is believed that this Premier League and global footballer wants a move to Jose Mourinho's Manchester United.
Doncaster-born Rose, who joined Spurs as a teenager in 2007, has made it clear that he intends to return to the north of England before he retires.