United recently exercised a one-year option to tie the defender to Old Trafford until 2019, but he might be used as a makeweight to capture Rose.
Per Nursey, Shaw arrived at the club from Southampton in a £27 million deal after impressing for Saints in the Premier League. The youngster soon broke into the England squad and was regarded as one of Europe's best developing talents.
However, his time at Old Trafford has seen him plagued by fitness issues, including a horrendous leg break in September 2015. Shaw's confidence was ruined after that major injury, and he is yet to return to his best form.
According to Samuel Luckhurst of the Manchester Evening News in July, United are prepared to persist with Shaw but will continue to monitor Rose's situation at Spurs.
Kyle Walker's big-money move to Manchester City could be viewed by Rose as the path for his career if United make a concrete offer for his services. Spurs manager Mauricio Pochettino worked with Shaw at Southampton and is well aware of the 22-year-old's potential if he needs a replacement at left-back.
Per Duncker, United could also turn to former Manchester City and Arsenal defender Bacary Sagna, who is without a club. The France international is hugely experienced and would offer Mourinho steady service from the bench.
The 34-year-old featured in 21 games for City last term in the Premier League and UEFA Champions League, according to WhoScored.com.
Sagna has played in English football since 2007, and United need coverage at right-back after allowing Timothy Fosu-Mensah to leave on loan to Crystal Palace.
Antonio Valencia has made the right-back position his own, but United lack knowledge in the position throughout the squad.
Sagna would be a great short-term option for United, as they battle for silverware domestically and in Europe.
Shaw deserves a year to prove himself to Mourinho, but the self-proclaimed Special One is notoriously impatient with players out of his starting XI.
The defender has shown flashes of brilliance when allowed to attack from deep, but Mourinho clearly questions Shaw's ability to fit into his system as a defensive-minded full-back.
Mourinho likes one full-back to sit while the other attacks, and Shaw has been exposed when tasked to do this.