The World Cup winner is expected to leave with Real Madrid and former club Juventus interested in signing him.
Saul, 25, has impressed for Atletico this season, helping them knock Liverpool out of the Champions League before the coronavirus pandemic stopped all football.
The playmaker has a massive £131million release clause but Spanish outlet AS claim Woodward is only willing to offer around £70m for his services.
Saul is currently in talks with Atletico over a new contract despite the COVID-19 crisis.
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His current deal runs out in June 2026 at the Wanda Metropolitano.
Saul has impressed this season, netting 4 times in 37 appearances across all competitions.
United boss Ole Gunnar Solskjaer thinks the dynamic Spaniard is in the best form of his career and would be the ideal fit for the reboot he is in charge of at Old Trafford.
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Saul also offers versatility as he can play on the wing as well as in central midfield.
If the Spaniard did join he would play alongside Scott McTominay – a player Nemanja Matic expects to star for years at Old Trafford.
“We are always talking, especially when he started to play in the first team, but now he's a proper senior player he plays like he’s had 10 years in the first team,” Matic said.
“With him, we don’t have any problems.”
“I think he will be a very important player for Manchester United in the next 10 years or more. He is the kind of player and person that you always want to have in the changing room.”
“[He’s] very professional, with a lot of quality in the legs,” he added.
“His mentality is what Manchester United needs to have, so I think that with any coach, in any club he will fit very well.
“I'm happy for him and United that they have a player like that in the team. He just needs to continue how he’s been working until now and I’m sure he will be one of the Manchester United legends one day.”
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