The 19-year-old joined Rangers in May after becoming less prominent at Liverpool under Jurgen Klopp than he was when Rodgers - who was sacked in October - was there.
Rossiter was on the fringes of Premier League action with the Reds and had already racked up a number of top-flight appearances for the Anfield club.
But the Liverpool-born midfielder's last start for the club was in a 1-1 Europa League draw with Sion just a few days before the Northern Irishman was dismissed.
Now, the two are very much rivals as Celtic manager Rodgers gears up for what promises to be a competitive season with Rossiter on the blue half of the Old Firm divide.
Asked by Rangers TV what his relationship with the Celtic boss is like, Rossiter said: "Obviously I owe a lot to Brendan. He brought me into the team [at Liverpool] and gave me a chance.
"Celtic are Rangers' main rivals and obviously you always want to beat them, but no I owe a lot to Brendan."
Rossiter was the first of nine signings made by Rangers boss Mark Warburton this summer, and the youngster will hope to meet Rodgers's Celtic in the first Old Firm clash of the season at Parkhead on September 10.