The Uruguayan was given a rapturous reception by the Reds supporters during Wednesday's win over Melbourne Victory, despite his recent angling for a transfer
Arsenal lodged a 49 million euro bid for the Uruguay international, while Real Madrid is heavily linked with a move for the striker who has made no secret of his desire to play Champions League football.
Suarez's attitude has alienated many Liverpool fans who stuck by him during the Patrice Evra racism row and Branislav Ivanovic biting incident, for which he is still serving a 10-match ban.
But the unsettled striker was still given a rapturous reception by Reds supporters during Wednesday's friendly win over Melbourne Victory and Rodgers believes this will play on the striker's mind.
"The support he has received from the supporters and the people of the city of Liverpool has been unrivalled," the former Swansea boss told reporters.
"The people have stood by him, like a son, and really looked after him. Whatever happens in the coming weeks that will be in his mind because it's something you can never forget."