The Daily Mirror says Rose remains United boss Jose Mourinho's first choice to strengthen the left-back position, but the Red Devils are not willing to pay Spurs' asking price for the England international.
Mourinho has predominantly used Ashley Young at left-back this season, while Luke Shaw has also featured sporadically following his return to fitness from a foot injury.
Tierney, 20, signed a new six-year contract at Celtic in October and is valued by the Glasgow club at around £25m.
Celtic manager Brendan Rodgers hailed Tierney as one of the top three full-backs in British football back in November and compared him favourably to Rose.
"I've worked with who I consider to be the two best full-backs in Britain, Danny Rose at Watford and Ryan Bertrand at Chelsea," Rodgers was quoted as saying by the Daily Mail. "Kieran is right up there with those boys.
"He's still very young, still has bits of improvement to make, but his quality, his will and determination and his focus for a young player is outstanding.
"He loves playing at Celtic. He's a special talent – but a boy who's very focused on improving."
Rose's future at Tottenham has been in doubt ever since he criticised the club's lack of ambition in the transfer market and announced he was open to offers elsewhere in a candid interview with the Sun back in August, with the club fining him two weeks' wages for the outburst.
The England international said he was "fuming" after being left out of Spurs' 2-0 defeat against the Gunners in November, but insisted that he had not fallen out with manager Mauricio Pochettino over the decision.
The 27-year-old has started five Premier League games for the north London club this season.