The Gunners made their move for the Scottish left back this summer, signing the 22-year-old on a long term contract.
Rodgers, who was manager at Celtic until early this year, said the Gunners' interest dated back to 2016.
"Peter [Lawwell, Celtic chief executive] had said to me - and he was very honest about it - there was a possibility of Kieran going to Arsenal, but there was no rush for Celtic to sell him, and there was a big offer at the time," Rodgers told reporters.
"So I spoke to Kieran, told him I wanted him to stay, to be a big part of what we were doing, and between him and the club, a new deal was agreed, and he stayed until he went in the summer.
"He's gone into Arsenal, a really competitive team and a huge club, and it takes that little bit of time to settle.
"He's gone from one big city to another, leaving behind his family and friends in Glasgow, so it's been a period of adaptation for him.
"I've got no doubt he will do very, very well in his time."