Jose Mourinho is set to take charge at Old Trafford and is ready to begin his recruitment drive as he prepares for life at the club.
United are believed to be on the lookout for significant reinforcements this summer as they bid to break back into the top four next season.
The Red Devils opted to sack Louis van Gaal on Monday after the club finished fifth and won the FA Cup.
Saul, 21, has attracted plenty of attention this season and is understood to have a £54m release clause in his contract at Atletico.
Now, talkSPORT is reporting that United are weighing up a move to bring the attacker to Old Trafford as Mourinho gears up for his first season at the club.
United will play in the Europa League next season after their fifth-placed finish.
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