Hill recently joined Scottish Premiership-bound Rangers after leaving Queens Park Rangers at the end of the 2015-16 campaign.
Rodgers was also appointed the Celtic manager this summer following the departure of Ronny Deila from Celtic Park.
The two recently bumped into each other in Glasgow, and Rangers defender Hill has revealed the conversation they had.
“I saw him in the West End the other night, which was funny,” Hill told The Daily Mail regarding Rodgers. “I was a bit like: 'What are people going to think if I'm shaking his hand?'
“But I have a lot of time for him. He was brilliant at Liverpool, which is my home team, and he's someone who seems approachable.
“He's very excited. The first thing he said to me was: 'I can't wait to get started.' It is intense up here but he will be ready for it."
Rangers won the Scottish Championship title last season with ease and will feature in the Scottish Premiership during the 2016-17 campaign.
Mark Warburton’s side have been quite busy in the summer transfer window, signing Hill, Joey Barton and Niko Kranjcar among others.
Celtic, though, will start next season as favourites for the league title and will have the strongest squad in entire Scotland.
The Bhoys’ main priority at the moment is to qualify for the group stage of the 2016-17 Champions League campaign.
Rangers will be expected by many to compete with Old Firm rivals Celtic for the Scottish Premiership title next season.