The Packers lost 20-16 to the Detroit Lions on Sunday at Lambeau Field.
The Packers had rallied from a 4-8 record by winning four straight games to set up a win-and-in situation against the Lions.
But Rodgers and Green Bay failed to uphold their end of the bargain, with the 39-year-old throwing a fourth-quarter interception on their last drive after the Lions had re-claimed the lead with Jamaal Williams' second rushing TD.
Rodgers is under contract for next season, but given his age and public comments, retirement has been a constant topic of discussion. The quarterback opted against being drawn on it in the immediate aftermath of the season-ending defeat.
"It's a little raw right now," Rodgers told reporters.
"It's just a little bit after the game, so I want to take the emotion out of it and have the conversation and see where the organisation is at and see how I feel after some time has passed.
"At some point, the carousel comes to a stop and it's time to get off, and I think you kind of know when that is.
"That's what needs to be contemplated. Is it time? Also, what’s the organisation doing? That's part of it as well. But the competitive fire is always going to be there. I don’t think that ever goes away."
Green Bay's loss was dotted with miscues including Rodgers' interception, along with a second-quarter Aaron Jones fumble and Quay Walker's inexplicable fourth-quarter ejection for pushing a Lions physician in the lead-up to Williams' game-winning TD.
The Packers missed the playoffs for the first time under head coach Matt LaFleur, who said: "Obviously very disappointed right now.
"To have an opportunity to go to the playoffs at home and not be able to get it done in that fashion is extremely disappointing.
"I do commend our locker room and our guys for their ability to overcome adversity throughout the course of the season.
"Ultimately, we dug ourselves too big of a hole and the margin for error in this league is so small, so if you don't capitalise on opportunities and make mistakes, then you're going to lose, especially against good football teams.
"You've got to give Detroit a tonne of credit, they came here and they beat us tonight. Certainly, like I said, [I am] extremely disappointed, but it's going to be a big offseason for us obviously to make sure we all improve.
"We've got to look inward to figure out what it is we need to do to get better, because the expectations here are very high and they should be. Really disappointed right now."