The 18-year-old made two Premier League appearances in the latter stages of the 2021-22 campaign, debuting against Chelsea at the end of April before featuring in his side's final game of the campaign at Crystal Palace.
Garnacho's impressive display against Rayo Vallecano in a pre-season friendly last weekend earned him a spot on the bench for Sunday's Premier League opener with Brighton & Hove Albion, and he was a late substitute for the Red Devils in their 2-1 defeat to the Potters.
Man United head coach Erik ten Hag is allegedly yet to decide whether to keep hold of the Argentina Under-20s international for the coming season or send him out on loan in order to continue his development.
According to The Mirror, more than 20 clubs in England, Spain, Germany, Italy and Scotland have expressed a desire to take the attacker on loan for the 2022-23 campaign.
The report claims that Garnacho's agent, Eugenio Ascari, met with Man United last week to discuss a possible short-term exit until January.
However, the 20-time English champions are yet to decide whether they are prepared to loan the teenager out due to his recent development.
Garnacho, who was named the Jimmy Murphy Young Player of the Year for 2021-22 due to his impressive form at youth level, joined Man United's Under-18s from Atletico Madrid in October 2020.
The attacker travelled with the Man United squad on their pre-season tour of Thailand and Australia but did not make a single appearance, instead having to wait until the friendly with Rayo last weekend to receive his first minutes of the summer.
"What you want is also a young player who deserves a chance," Ten Hag told MUTV after the Rayo clash. "We want to give them that chance and he shows himself during pre-season [training] and that is why he came on [started]. He played pretty well and that's hopeful. He is just a teenager but his performance was really good."
The South American starred at Under-18 and Under-23 level during the 2021-22 campaign, including four goals and one assist in Premier League 2, and he netted two late goals as the Red Devils won the FA Youth Cup.
Garnacho was introduced in the 90th minute of Sunday's clash with Brighton but was unable to prevent his side from suffering a narrow defeat in their season opener.
Pascal Gross scored twice in the space of nine first-half minutes to hand Graham Potter's side a two-goal advantage at Old Trafford, which proved enough to secure the points, with Man United's goal in the 68th minute only a consolation.
Cristiano Ronaldo was introduced in the early stages of the second half against the Seagulls, but the 37-year-old, who continues to be linked with a summer exit, found it difficult to make his mark.
Lisandro Martinez and Christian Eriksen both started for Man United in their season opener, while fellow new arrival Tyrell Malacia was a late substitute.