Manchester United are interested in signing Sanchez despite the player long being tipped to end up across town at Manchester City.
The 29-year-old’s contract at the Emirates expires at the end of the season, with Arsenal ready to cash in this month to avoid losing him for free.
Express Sport understands the Red Devils have launched a £25million bid for Sanchez, and are also prepared to include Henrikh Mkhitaryan in any potential deal.
And the Daily Mail say that, in order to ensure Manchester United beat City to the forward’s signature, Mourinho has made a promise to club chief Woodward.
Vice-chairman Woodward is the man responsible for bringing players to Old Trafford.
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During his time in the job, he has smashed the club’s transfer record three times and the world record once.
Yet Mourinho, the Daily Mail say, has assured Woodward he will not be greedy should the club conclude a deal for Sanchez.
As a result, he is even prepared to give up on Gareth Bale and Antoine Griezmann.
And any deal for Sanchez would also see the club give up on their dream of bringing Cristiano Ronaldo back to the Premier League.
Ronaldo spent six years at the club between 2003 and 2009 before signing for Real Madrid.
But nine years on from that deal and the 32-year-old’s future is uncertain.
United, meanwhile, may opt for shrewder transfers at the end of the season if Sanchez arrives.
The club are likely to have around £150million to spend and Sanchez’s Arsenal team-mate, Mesut Ozil, is a target on a free transfer.