According to David McDonnell of the Daily Mirror, the Red Devils are not willing to part with the £50 million that would be needed to sign Rose and are considering alternatives as a result.
Tierney is one of those, and McDonnell noted the Scotland international "would cost around £25 million and has the added bonus of also being able to play at right-back."
United boss Jose Mourinho is said to be keen to add another left-back to his squad despite Luke Shaw's upturn in form of late. As football writer Jordan Clarke noted, after a tough beginning to the season, the England international has thrived in recent matches:
Even so, Shaw is the only natural left-back in the squad, with Ashley Young, Matteo Darmian and Daley Blind all used as makeshift options in the role in 2017-18.
As a result, the pursuit of someone like Rose would make sense for United, as he's proved himself to be one of the Premier League's best players at the position. The Tottenham man has hinted in the past he may be keen on a move away from London too, stating in an interview with The Sun's Dave Kidd he's eager to play "up north" again.
This season, Rose was slow to get going following a long-term injury layoff. However, he's looked back to a level approaching his best again recently and is providing stiff competition to Ben Davies for a starting spot.
Jack Pitt-Brooke of The Independent suggested Rose still has the edge on his team-mate:
It'd be understandable if United were hesitant about paying so much for a player who missed the majority of 2017 because of injury. Tierney represents a fascinating contingency plan.
The Celtic man has been exceptional for Brendan Rodgers' side in recent years, tearing up and down the left flank with menacing intent. Tierney is tenacious in his defending, has exceptional energy reserves and has shown he can make an impression going forward too.
Although he's not as refined as Rose or proven in the Premier League, Tierney has all the tools to blossom into a wonderful left-back. Scouted Football rates him highly:
Scouted Football @ScoutedFtbl
Kieran recovers from a ruptrued cruciate ligament in three months, just as Celtic return from their winter break. He started in a league win vs St. Johnstone, setting the tone for an extraordinary 12 months. For club and country, he'd go on to miss just 5 games in 2017. https://t.co/y1Ou7yCtuz
Tierney, just 20, needs to continue playing to make improvements in his game, and he's working under a fine coach in Rodgers.
A move to United, where he would potentially be down the pecking order, may halt that progression, especially if he were to join during the January transfer window.
With Shaw performing admirably, the Red Devils' need for a new left-back appears to have diminished. However, if United could get Rose in and the defender were able to rediscover the form he showcased early in the 2016-17 season, he would add another dimension to the side.