The Daily Mirror says Spurs boss Mauricio Pochettino is resigned to losing Rose and is drawing up a list of potential replacements for the England international.
Manchester City boss Pep Guardiola is also interested in Rose, but United are reported to be in pole position to land the 26-year-old due to Tottenham's interest in Luke Shaw as a replacement.
The report comes days after Rose said some Spurs players "might fancy a change" and leave the club in the summer after another trophy-less season.
Right-back Kyle Walker has also been linked with a move away from north London, with City reportedly keen on signing him.
"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 last week.
"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.
"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'."