The Scotsman, who replaced Sir Alex Ferguson at Old Trafford last summer, has come under increasing pressure due to United's poor season.
The Red Devils recorded their 10th Premier League defeat of the campaign after Manchester City secured a 3-0 win at Old Trafford on Tuesday night.
Gillespie wrote on Twitter: "I'm fast coming to the conclusion that the David Moyes experiment needs to come to an end sooner as opposed to later.
"The United players collectively are better then the performances we have seen so sadly the coaching has gone amiss. Therefore how good is he
"In betting terms Mourinho was a safe bet and Moyes was risk due to experience. Unfortuneatly risk seldom pays off and I should know!!!"
United are currently seventh in the league table, 12 points adrift of a Champions League spot.