“Rodgers out, Rafa in”, read the message in a stunt organised by a campaign group called RodgersOutClub.
Rodgers, appointed in May 2012, is the first Liverpool manager in more than half a century not to win a trophy in his first three seasons at the club.
Many fans and former players including Jamie Carragher condemned the banner.
Earlier on Saturday, Carragher tweeted: “About to leave for Anfield. Hope this Rodgers banner supposedly flying over the ground is called off as it’s embarrassing to club and fans.”
Rodgers, 42, who took the Reds to the brink of the Premier League title last season only to lose out to Manchester City, believes he is still the right man to take the club forward.
He also insists the club’s American owners, Fenway Sports Group, will support him with marquee signings this summer.
Former Reds manager Rafa Benitez, who led the club to Champions League success in 2005, is expected to leave Serie A side Napoli at the end of the season.