The 28-year-old has been captain of the red ball team for the past two-and-a-half years, but was replaced in that role on the eve of the new season, which starts on Thursday.
Rossington's contract at Northants expires at the end of 2022.
“It’s disappointing Adam doesn’t see his future with Northamptonshire, but everyone at the club truly wishes him well for the future," said chief executive Ray Payne.
"We thank him for his fantastic service over the past eight years and as captain most recently.”
Rossington, who joined the club in the summer of 2014, took over from Alex Wakely as captain midway through the 2019 campaign.
He played a major role in turning the fortunes of the club around, so much so that the County rose from the foot of the table to finish second and claim promotion to the top flight.
But now will not be in charge when the reward for that promotion is claimed this summer.
“Adam did a brilliant job as captain after taking over in difficult circumstances in 2019, and the Cricket Committee had Adam as a central part of our plans moving forward," added Payne.
“Conversations around keeping him here had been ongoing since the end of last season, so it’s a shame he won’t be with the squad as they take that place in division one.”
Rossington initially joined Northants on loan from Middlesex midway through the 2014 season, and signed for the club on a permanent basis the following winter.
In all, the wicket-keeper batsman made 192 appearances for the club, and was part of the Steelbacks team that won the T20 Blast in 2016.