Bodden, who spent Monday in Houston visiting the Texans, had also been keeping open a dialogue with the Patriots. While an agreement has been reached with New England, Bodden's agent, Alvin Keels, cautioned that no contract has been signed at this point.
The source told Schefter that contract details are being worked out.
"Reports of Leigh Bodden re-signing with the Patriots are not true and premature at best," Keels wrote on his Twitter account. "Leigh is still evaluating his options. Leigh has always indicated his desire to return to New England, but nothing is official and probably will not be until [Tuesday] at the earliest."
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.
Keels had also been speaking with the Steelers and Seahawks regarding the seven-year veteran Bodden, who spent 2008 in Detroit after his first five seasons in Cleveland.
"It's kind of nerve-wracking," Bodden said on Sunday of visiting Houston. "It is what it is. It's part of the business. Hopefully I can get things done with the Texans and just move on."
The Patriots had considered it important to retain Bodden, who finished with 55 tackles last season and was one of the team's most consistent players on defense. His 82-percent playtime total ranking third among all defenders, while his toughness was exhibited as he played through injury.
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.
Now the sturdy Bodden (6-foot-1, 193 pounds) is in line to receive the largest contract of his seven-year career.
Meanwhile, the Patriots retain their best cornerback and a solid locker-room presence. Had Bodden signed elsewhere, the Patriots' need at the position would have been high, with second-year man Darius Butler, 14-year veteran Shawn Springs and third-year players Jonathan Wilhite and Terrence Wheatley topping the depth chart.
Keeping Bodden in the mix on the right side is a key piece, giving the Patriots the flexibility to address other needs.
The Patriots don't appear to be done dealing, either. Rams free-agent tight end Daniel Fells spent the day in New England, and the sides were discussing a deal, a source told Schefter.
ESPNBoston.com's Mike Reiss contributed to this report.