The deal includes $10 million in guaranteed money and $14 million over the first two years of the deal, the source said.
Patriots owner Robert Kraft confirmed an agreement had been reached in an interview on Boston sports radio station WEEI, and Bodden's agent, Alvin Keels, did as well via Twitter.
Schefter first reported the sides had an agreement on Monday night.
Bodden, who spent Monday in Houston visiting the Texans, had been keeping open a dialogue with the Patriots. Bodden's preference was to remain in New England, but financial considerations also played a part in his decision. Having entered the NFL as a rookie free agent out of Duquesne, he had never received a big payday on the open market.
The 6-foot-1 Bodden was the Patriots' top cornerback in 2009, playing in 15 games while leading the team with 17 passes defended. He tied for the team lead with five interceptions, three of which came against Jets quarterback Mark Sanchez.
Keeping Bodden in the mix on the right side is a key piece, giving the Patriots the flexibility to address other needs.
Kraft also confirmed during the radio interview the team had an agreement with free-agent linebacker Marques Murrell.
Murrell, whose primary contributions have come on special teams but who was known more for his pass-rushing skills at Appalachian State, has been with the Jets over the last three seasons. He has appeared in 26 regular-season games (zero starts).
The 6-foot-2, 250-pound Murrell was a restricted free agent. Since the Jets did not tender him a contract, he became an unrestricted free agent.
The younger brother of former NFL running back Adrian Murrell, he entered the NFL in 2007 with the Eagles as an undrafted free agent.
At Appalachian State, he recorded 36 sacks and set a school record with 18 forced fumbles. Appalachian State won two Division I-AA national championships (now FCS) during his time with the team.
In 2009, the 24-year-old Murrell made headlines for the wrong reasons when he was suspended one game by Jets coach Rex Ryan for an on-field scuffle with Bills tight end Derek Fine.
Also Tuesday,former Patriots defensive lineman Jarvis Green signed a four-year contract with the Denver Broncos. The deal is worth up to $20 million, and will pay Green $7 million over the first two years between bonuses and base salaries, league sources told Schefter.
"It's unfortunate in this business you can't keep everyone you want to keep," Kraft said.
Green, 31, who spent the last eight seasons with the Patriots, was the team's starter at right defensive end in 2009. But he was most effective over his career as an interior sub rusher, with 28 career regular-season sacks.
Green's contract with Denver highlights a significant void the Patriots must fill at right defensive end. While Green isn't best suited to be a 16-game starter at that spot, he has been a valuable reserve who was effective when called upon to start in stretches.
ESPNBoston.com Patriots reporter Mike Reiss and information from The Associated Press contributed to this report.