The Daily Mirror says United boss Jose Mourinho remains unconvinced by his current options at left-back and could sanction a move for England international Rose.
Serie A club Inter have also kept tabs on Rose since last season and could launch a bid before the transfer window closes.
Rose made 18 league appearances for Spurs last season, scoring two goals. He has not played for the club since suffering a knee injury in January, but is expected to return to action before the end of the month.
The England defender said in May that some Spurs players "might fancy a change" and leave the club in the summer after another trophy-less season.
"There is only so much the club can do [to keep a player]; a lot of it has got to be down to the individual," Rose told the BBC.
"Someone like Harry Kane, I can tell you that his mindset will be 'I'm staying at Tottenham, I'm going to break every single record, I'm going to captain this club into the new stadium' so when you've got a player like him with that mindset I don't think Tottenham have to worry.
"But it depends where the individual is in the stage of his career. He might feel he has done enough at Tottenham and might fancy a change or might feel Tottenham is the nearly club that nearly wins the league and might choose to move on," he added.
"The club can only try and make a player see that they are going in the right direction and 'if you stick with us it won't be long before we are winning something'."
Tottenham are the only side in England's top four divisions yet to make a summer signing.