The plane bearing the message was spotted during the Reds' 2-1 win over Chris Ramsey's side on Saturday, with Rodgers' side currently four points behind fourth-placed Manchester United in the race to qualify for the Champions League.
Rodgers brushed off the stunt - with fans calling for the Napoli manager, who won the Champions League in 2005 during his spell in charge of the Reds, to replace him - but admits that "football has a short memory", with Liverpool having finished second behind eventual champions Manchester City last season.
“I thought it was Rafa’s agent in it,” he told reporters.
“We sat here last year having nearly won the league and now we have planes flying over so you can do nothing about it.
“I’m very fortunate and privileged to work here at Liverpool and as I said before there will be many names linked with the job because of the size and scale of the job to be undertaken here.
“I’m hugely honoured every single day of my life, and I will continue to be, however long it lasts.”
Rodgers also hailed captain Steven Gerrard, who recovered from the disappointment of missing a penalty to score the winning goal in his penultimate game at Anfield prior to his move to LA Galaxy this summer.
"There's a very, very small group of players in that bracket, where in the big games they score the big goals and make the big contributions," Rodgers added.
"He's [Gerrard] done that, everyone has seen that throughout his entire career at Liverpool.
"He missed the penalty and sometimes you think it may go against us, but there is no-one more determined to score and make up for it than Steven.
"It was a very, very good header. He was very determined, especially after the penalty.
"You sometimes find with a lot of players that when they miss a big penalty like that, normally their game goes and their confidence goes.
"But he's the total opposite. He's one of a unique group of players that whenever they need to respond, they normally do. It was an outstanding header and obviously gets us the win."