Whites Academy graduate Phillips has ended a 12-year association with his boyhood club by signing for City on a six-year-deal.
City are paying £42m for the midfielder - rising to £45m with add-ons - but Leeds believe the deal will be worth closer to £50m to them for a player who joined the Whites when he was just 14 years old.
Cooper was coming to the end of his first season at Leeds when Phillips made his United debut as an 18-year-old in the Championship clash at Wolves of April 2015.
The duo went on to fire United to promotion five years later and Cooper is confident that Phillips will continue to produce the goods upon his move to City.
The Whites skipper also urged his former team mate to never change who he was as a person having always won hearts during his time with the Whites.
Leading the tributes to Phillips on social media, Cooper wrote: "Like a proud father.
"You have earned this chance and I have no doubt you will deliver like you always did for us.
"An unbelievable player but I'm more proud of the man you have become!
"Never change mate. Look after him.”
Cooper’s post came in response to an emotional message of farewell from Phillips which gained almost 100,000 likes on Instagram within a few hours.