Wenger was subjected to an insulting chant urging him to sit down during Arsenal's 2-1 Barclays Premier League defeat at Old Trafford on Saturday.
It is understood Gill is appalled by the treatment dished out to one of the most distinguished managers in the game, says the Daily Mail. Gill is to attend a fans' forum next month - the club hold three per season - and will make the Wenger abuse a priority for discussion.
United had already decided to act before receiving a complaint from an Arsenal supporters' group.
United spokesman Philip Townsend said: "We've gone on the record several times about this disgusting chant. We don't condone it and have appealed to fans in the past - through supporters' groups, the matchday programme and MUTV, but to no avail. There are many chants that opposing fans find objectionable, and this is certainly one to which all decent supporters should object."