The Red Devils have allegedly agreed to sign the Denmark international on a three-year contract, and it was initially thought that the 30-year-old's signature would be secured towards the start of this week, which would allow him to board the plane on Friday for his side's pre-season trip to Bangkok.
However, according to BBC Sport, Man United are yet to arrange a medical for the attacker, while there are still some small details that need to be addressed before the move is completed.
As a result, the report claims that Eriksen is unlikely to make the trip to Thailand to face Liverpool and could instead travel straight to Australia, with the Red Devils facing Melbourne Victory, Crystal Palace and Aston Villa in the country before returning to Europe.
Eriksen's debut for the club could therefore come against Melbourne Victory on July 15, while he is set to feature in the matches on European soil against Atletico Madrid and Rayo Vallecano.
Brentford had allegedly been hopeful of convincing the playmaker to sign a fresh deal at the club, but the Bees have seemingly failed in their bid to keep hold of him.
Eriksen had an implantable cardioverter defibrillator device fitted after suffering a cardiac arrest while representing his country at Euro 2020, which saw him leave Inter Milan.
The Dane moved to Brentford on a free transfer in January and impressed for Thomas Frank's side, scoring once and provided four assists in 11 Premier League appearances in the second half of last season.
Tottenham Hotspur, Everton, Newcastle United and Leicester City are all believed to have shown an interest in signing Eriksen on a free transfer this summer.
However, Man United appear to have won the race for a player who will boost an attack that has been weakened by the departures of Paul Pogba, Juan Mata, Jesse Lingard and Edinson Cavani this summer.
"I have different offers and options, which we are considering, and then we make a decision. I would love to play Champions League football again. I know how much fun it is, but it isn't essential for me," Eriksen told Viaplay when questioned on his future.
Eriksen previously turned out for Tottenham between 2013 and 2020, making 305 appearances for the English club in all competitions, scoring 69 goals and registering 90 assists.
Providing that the attacker passes his upcoming medical and does not suffer any fitness problems during pre-season, he could make his Premier League debut for the 20-time English champions against Brighton & Hove Albion on August 7.
Erik ten Hag's side will also take on Brentford, Liverpool and Southampton in the opening month of the new Premier League season.
Tyrell Malacia became the club's first signing of the summer on Tuesday, while a deal for Ajax centre-back Lisandro Martinez has also allegedly now been agreed.