Sanchez has been linked with a move to Manchester City for several months and almost joined the current Premier league leaders last summer, but United manager Jose Mourinho is now keen to get one over on rival Pep Guardiola.
The Red Devils have made an offer in the region of £25m for Sanchez and are also willing to include out-of-form Armenian playmaker Henrikh Mkhitaryan as part of the deal.
Sanchez has already been offered £250,000-a-week by City, which the Chilean has agreed to but the two clubs have failed to agree on a price for the forward.
City are not willing to pay over £20m for Sanchez, who could be available for free in the summer when his contract with the north London club expires.
United are hoping that the extra £5m in cash plus Mkhitaryan will be enough to persuade the Gunners to enter negotiations with them rather than their noisy neighbours.
United are expected to offer Sanchez a better personal deal than City, but the player is understood to be keen to reunite with former boss Guardiola.
Signing Sanchez would be a huge confidence boost for United, as the Chilean will be able to feature in the UEFA Champions League.