Rose is back from a long-term injury, but the England international defender is not guaranteed a place in the Spurs starting lineup.
The 27-year-old left-back was left out of the Tottenham team for the North London derby against Arsenal, as well as against West Bromwich Albion in the Premier League at the weekend.
Former United and Everton star Neville, who now works as a pundit for Sky Sports, does not believe that Rose will stay at Tottenham.
Former Chelsea midfielder Andy Townsend has also suggested that Rose will not be at Tottenham.
The Daily Star quotes Townsend as saying about Rose’s future: "I've got a feeling he'll move on. I think he'll go.
Something’s clearly rattled Danny, when he said what he said in the press and he got ticked off for that.
“He's obviously had his injury and Ben Davies has improved rapidly as a player. I can see him ending up at [Manchester] United."
Rose is one of the best left-backs in the Premier League, and if United sign him, then they would be getting a player who would slot straight into Jose Mourinho’s team.