The Red Devils were touted as a potential option for Sanchez this month as Jose Mourinho looks to improve his attacking options at Old Trafford.
Reports on Thursday suggested Manchester United are willing to send Henrikh Mkhitaryan to north London as part of a potential deal to sign Sanchez.
The Chile international has been heavily linked with a move to Premier League leaders Manchester City over the past 12 months or so.
The 29-year-old has less than six months remaining on his current Arsenal deal to raise the possibility of Sanchez leaving for nothing at the end of the campaign.
The South American worked under Manchester City boss Pep Guardiola during a three-season spell at Barcelona before the forward completed a £35m move to Arsenal in 2014.
BBC Sport reporter Stone rated Manchester United’s chances of signing Sanchez as “slim”, despite the possibility of using Mkhitaryan as part of a swap deal.
“The chances of Man United signing Sanchez are slim because he wants to play for Pep Guardiola again,” Stone told BBC Radio 5 Live.
“Guardiola wants him to play for Man City. Man United are saying nothing about this. They are at pains to stress that the fact they are saying nothing isn’t them knocking it down.”
Manchester United have been linked with a move to sign World Cup winner Mesut Ozil over the past couple of months.
The Red Devils are in second place in the Premier League table but 15 points adrift of leaders Manchester City.
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