Scott McTominay (left) made his Manchester United debut in 2017 while Harry Maguire (right) moved to Old Trafford for £80m from Leicester City in 2019
The Hammers have yet to make a signing this summer despite selling Declan Rice to Arsenal in a deal that could end up being worth £105m.
They are also in advanced talks with Ajax over 25-year-old Mexico midfielder Edson Alvarez.
Meanwhile, Italy striker Gianluca Scamacca, 24, has left the club to join Atalanta for an undisclosed fee.
After Saturday's 4-0 defeat by Bayer Leverkusen, there is greater momentum behind the Hammers' recruitment drive.
Whether West Ham can reach either the fee or salary requirements to conclude a deal for the Manchester United pair is open to question.
It is understood United would be open to a deal, but both England defender Maguire, 30, and Scotland midfielder McTominay, 26, are part of Erik ten Hag's squad and the club know they would need replacements for either if they were sold.
Scamacca spent one season with the Hammers having joined on a five-year contract from Sassuolo for £30.5m last July.
He made 27 appearances and scored eight goals, including five strikes in their victorious Europa Conference League campaign.
-- Courtesy of BBC Sport