Rodgers joined the Reds in 2012 and has less than two years remaining on his current deal.
According to The Mirror, Liverpool do not want to take any chances and are keen to tie the Northern Irishman to a long-term deal.
Rodgers won the manager of the month award for August for a bright start to the Premier League by his side.
Liverpool are said to be aware that other clubs are monitoring his situation and the likes of Arsenal, Manchester City and Tottenham Hotspur could move for Rodgers in the future.
The FA are also said to be considering him should Roy Hodgson fail to qualify England for the World Cup in 2014.