As reported by Goal, Minamino was one of a number of players United have been tracking ahead of the January window and was placed on a shortlist of potential transfer targets.
Liverpool paid the Japan international's £7.25 million ($9.6m) release clause to bring him to Anfield from Red Bull Salzburg, with United under the impression he would cost far more.
The Red Devils, whose transfer business is led by Woodward, were unaware of Minamino's release clause.
United are keen to move on from their failure to secure the Salzburg star, and the club's squad are of the belief that two new signings will be joining them following the turn of the year.
A creative midfielder is high on Ole Gunnar Solskjaer's shopping list, with Leicester City midfielder James Maddison seen as a potential option, though Foxes boss Brendan Rodgers has insisted none of his key players will be leaving the King Power Stadium in January.
United made a move late in the summer transfer window to sign Eriksen but failed to reach a deal. They will, however, have been boosted by an interview given by Spurs chairman Daniel Levy , who told the Evening Standard that he would not block Eriksen from joining a rival Premier League team.
It is understood Levy is reluctant to allow Eriksen to leave for free in the summer and as such is keen to secure a fee for the former Ajax star in the coming weeks.
United's need for a midfielder comes amid increased scrutiny regarding the future of Paul Pogba.
The World Cup-winner missed Wednesday's Carabao Cup win over Colchester United through illness as his return from an ankle injury, which has seen him sidelined since the end of September, was again delayed.
Solskjaer, meanwhile, is also keen to reinforce his forward line in January.
Juventus outcast Mario Mandzukic had been the club's prime target, but their attention has now turned to Minamino's Salzburg team-mate Erling Braut Haaland.
Solskjaer - who previously managed the 19-year-old during his time at Molde - met with the striker in Austria last week to discuss a potential move to the Theatre of Dreams.
Haaland, meanwhile, has visited both Borussia Dortmund and RB Leipzig in the last 10 days, with the Bundesliga sides also keen on a player who has scored 28 goals in 22 appearances for Salzburg this season.
Jadon Sancho is another who has been on United's wanted list and the club would be willing to pay as much as €140m (£119m/$155m) to land the England international.
Goal understands that Chelsea are, however, increasingly confident of landing the 19-year-old ahead of United, either in January or next summer, with Sancho needing to be convinced that the Red Devils are capable of pushing for a Champions League place.
Solskjaer's side have three fixtures to play before the January window opens, starting with Sunday's trip to face the Premier League's bottom side, Watford.
They then host Newcastle on Boxing Day before making the trip to Turf Moor to take on Burnley in their final match of 2019.