Robbie Brady scores for Ireland against Italy
The Hammers' Premier League rivals Leicester City and Southampton are also linked with moves for the Republic of Ireland star, though United topped the list of bookies' favourites on Tuesday - despite there being no reported interest from Slaven Bilic's team.
Despite making 36 appearances and scoring three goals in the league last term - two of which came against West Ham - Brady was relegated for the second season running.
Brady celebrates scoring against West Ham
And following a string of impressive displays at Euro 2016, it is unlikely the 24-year-old will begin the new campaign a Championship player.
West Ham have already added Valencia winger Sofiane Feghouli to their ranks this summer, but are said to be chasing additional signings in Brady's position - notably Besiktas man Gokhan Tore (Sky Sports).
Champions Leicester could prove difficult opposition in the Hammers' alleged pursuit, with the versatile left-sider having European Cup football to look forward to if he moves to the King Power Stadium.
Leicester City players prepare to lift Premier League trophy
And Southampton might also be an attractive proposition for the former Hull City star, having already acquired his Carrow Road teammate Nathan Redmond this summer.
Brady again on target against West Ham - this time for Hull