The high flying Premier League side do not want to lose their talented Northern Irish boss.
Rodgers has overseen a turnaround in form since taking over from Claude Puel last season, with Leicester currently second in the Premier League.
The Telegraph says Rodgers will be offered a significant extension to his contract, which should give him a sizeable pay rise.
He is among the candidates for the Arsenal job, but Leicester hope signing a new deal will end any doubts regarding his future.
Rodgers still has roughly 2.5 years left on his contract that he signed last season, while it contains a £14 million release clause.