City had been the favourites to sign Sanchez but chose to walk away after it became apparent that the cost of completing the transfer this month would exceed the £60m fee they agreed with Arsenal for the Chilean last summer only for the deal to collapse on deadline day.
United have offered Henrikh Mkhitaryan, who would prefer a return to Borussia Dortmund, as a makeweight as they seek to match Arsenal’s £35m valuation of Sanchez.
They are also prepared to meet Sanchez’s demand for a signing-on fee of up to £20m in addition to footing a potential £10m bill for the striker’s agent, Fernando Felicevich, which would value the overall deal at £65m for a 29-year-old who is out of contract at the end of the season.
That figure does not include the reputed £400,000-a-week wages, including bonuses, that would make Sanchez the highest-paid player in the Premier League.
United remain the front-runners but they face competition from Chelsea, who have yet to be deterred by the huge costs involved in trying to sign Sanchez.