Despite struggling in the domestic league in his first season, Mourinho guided the Red Devils to win the Community Shield, the League Cup and the Europa League.
“I hope so [lead the club to success] because it’s not easy to repeat what [former United manager] Sir Alex did there,” Essien told the Manchester Evening News.
“I am sure deep down, Jose will be thinking 'if I can even do half of what Sir Alex did for Manchester United then I would have done well'.
“Three trophies in his first season is not a bad start.”
Essien might have played out a season-long loan under Mourinho in 2012-13 but the one-year deal at Real Madrid stands far from their biggest connection.
The duo spent three seasons together at Chelsea between 2005 and 2007, winning the Premier League, the League Cup and the FA Cup.
“He was my father and I was his son, that’s as personal as it gets," Essien added.
“Some of my personal memories of him will always remain with me.
“He’s one of the best in the business in terms of preparing for games and preparing for the season as a whole. Everything is planned in advance.
“Also, as a player you knew what he expected of you and where you stood with him.
“He says it as it is and never shies away from criticising you if he is not happy with you. At the same time, he is the first to praise you when you do well.”
Essien currently plays for Indonesian side Persib Bandung.