"We anticipated bigger offers in free agency, but Aaron wanted to stay with the Packers," Rosenhaus told ESPN.
The Packers decided not to use their franchise tag on Jones before the deadline to do so last Tuesday and instead continued talks on a long-term contract.
Jones tweeted "let's run it back" on Sunday.
He also said, "I'm just glad I could keep playing where I started my career" on Instagram Live.
Jones, a fifth-round pick by the Packers out of UTEP in 2017, ranked fourth in the NFL last season with 1,104 rushing yards despite missing two games with a calf injury. He made his first Pro Bowl -- becoming the lowest-drafted Packers running back since Dorsey Levens (also a fifth-round pick) in 1997 to do so.
It was Jones' second straight 1,000-yard rushing season. He rushed for 1,084 yards and led the NFL with 19 touchdowns in 2019. Including the playoffs that season, he scored 23 times, the most for a season in team history.
Jones, 26, is one of only two players in NFL history to post 3,000-plus yards rushing (3,364) and 35-plus rushing touchdowns (37), with an average of 5-plus yards per carry (5.2) in their first four seasons. Jim Brown is the other.
The Packers had been in contract talks with Jones since February 2020. Late last season, Jones, who became frustrated with the lack of guaranteed money the team was offering, changed agents and hired Rosenhaus. At the time, the Packers had offered Jones a deal that would have paid him among the top-five running backs in the NFL in terms of average per year, a source told ESPN, but not in guaranteed money.