Van Gaal, who was placed in charge at Old Trafford in May, has been nominated after guiding the Netherlands to third place in the Brazil World Cup in the summer.
Mourinho, who failed to win silverware during his first season back at Chelsea in the 2013-14 campaign, is part of the 10-man shortlist, alongside Champions League-winning Real Madrid coach Carlo Ancelotti.
Germany manager Joachim Low has also made it onto the list after the nation were crowned world champions in July, while Manchester City's Manuel Pellegrini and Bayern Munich's Pep Guardiola are also in contention.
There is also a place for United States national manager Jurgen Klinsmann.
Coach of the Year shortlist:
Carlo Ancelotti (Real Madrid)
Antonio Conte (Juventus/Italy)
Pep Guardiola (Bayern Munich)
Jurgen Klinsmann (United States)
Joachim Low (Germany)
Jose Mourinho (Chelsea)
Manuel Pellegrini (Manchester City)
Alejandro Sabella (Argentina)
Diego Simeone (Atletico Madrid)
Louis van Gaal (Netherlands/Manchester United)