Alonso is available on a free transfer this summer, with the Spaniard leaving Barcelona at the end of June, and he could allegedly be offered the chance to return to the Premier League.
According to The Athletic, senior figures at Man United have discussed the possibility of bringing in Alonso, who would represent a low-cost experienced option.
Man United have made four signings this summer, with Joshua Zirkzee and Leny Yoro joining last month, before deals for Matthijs de Ligt and Noussair Mazraoui were confirmed on Tuesday.
Mazraoui is capable of playing left-back, while the same can also be said for Diogo Dalot, but the Red Devils are said to be seriously considering signing a specialist in that area of the field.
Alonso 'emerges as genuine target' for Man United
Shaw has not played for Man United since the middle of February due to a muscular problem, but he was able to feature for England in the latter stages of this summer's European Championship.
The ex-Southampton youngster is facing more time on the sidelines, though, with Man United revealing on Tuesday that a calf problem will keep him out until the middle of September.
Tyrell Malacia missed the entirety of the 2023-24 campaign with a serious knee injury, and it is still unclear when the Dutchman could return, so the Red Devils again have big problems at left-back.
Lisandro Martinez started in that position against Manchester City in the Community Shield on Sunday, while as mentioned, Dalot and Mazaraoui are capable of featuring in that area; Harry Amass, 17, is also an option, but he is yet to make his senior debut for the 20-time English champions.
Would Alonso be a good signing for Man United?
Alonso certainly knows what it takes to play in the Premier League, representing Bolton Wanderers between 2010 and 2013, including nine appearances in England's top flight.
The Spaniard then played for Sunderland on loan in the Premier League during the 2013-14 campaign, before moving from Fiorentina to Chelsea in 2016, and he made 212 appearances for the Blues in all competitions, including 154 outings in the top flight of English football.
Alonso won the Premier League, FA Cup, Champions League, Europa League, UEFA Super Cup and Club World Cup during his time at Stamford Bridge before moving to Barcelona on a free transfer in 2022.
The 33-year-old was a regular for Barcelona in 2022-23, making 37 appearances, helping the club to win the La Liga title, but he only featured eight times in all competitions last term.
Alonso's lack of football last season is a concern, but he still has something left to offer at the top level, and bringing him in as cover at Old Trafford could prove to be a smart piece of business by the Red Devils.