For now, though, the team and 35-year-old player are going their separate ways, the source said.
"I'll have teams calling me, and they won't be [bad] teams," Springs told The Boston Globe. "I'm cool. I would come back to New England. I wish them the best. I'm going to miss those guys."
Springs told the Globe that he had arthroscopic knee surgery in January, though he stressed that the injury was not the reason why he was released.
"It's not a knee issue," he told the newspaper.
However, a league source told ESPNBoston.com that Springs failed a recent physical, though it is unclear whether this played a factor in his release.
In signing a three-year deal last offseason, Springs landed a $2.7 million signing bonus and a $1.75 million base salary. So for $4.55 million, the Patriots received 12 games, 39 tackles, one interception and four passes defended.
Springs was due to earn $2.25 million this year, the ninth-richest salary among base salaries of Patriots players.
The selection of cornerback Devin McCourty in the first round of April's draft might have had something to do with releasing Springs at this time.
"It's almost like [coach Bill Belichick] cut me, and he wants to see what he's got without distractions," Springs told The Globe. "He wants to give them a shot. It's hard to give them a shot when you have the presence of a guy like me around. He's going to give you all the chances not to mess it up. It helps your psyche. You don't have to compete. Just do your job.
"Now, you don't have to worry about the pressure of guys competing with me and messing up your confidence as a young guy, feeling like, 'How am I going to compete with this guy?' "
Springs began his career in 1997 with the Seahawks and made the Pro Bowl in 1998. In 2004, he left Seattle for Washington and spent five seasons there before signing a three-year, $10.5 million contract with the Patriots prior to last season.
The Patriots also announced that they have signed offensive lineman Ted Larsen, a sixth-round pick out of North Carolina State. He received a four-year contract worth $1.87 million, his agent told ESPNBoston.com's Mike Reiss.
Adam Schefter is ESPN's NFL Insider. Information from ESPNBoston.com's Mike Reiss and The Associated Press contributed to this report.