The free agent agreed to contract terms with the Giants on Monday, a source with knowledge of the negotiations told The Associated Press on condition of anonymity because the deal has not been announced.
Contract terms were not immediately available.
Drew Rosenhaus, Bradshaw's agent, did not return an email from The AP seeking comment. The Giants refused to comment because the deal was not signed.
Bradshaw took over as the Giants' starting halfback last season and rushed for a career-best 1,235 yards and eight touchdowns. He had problems with fumbles, and Brandon Jacobs started the next five games. Bradshaw started the season finale against Washington.
Bradshaw lost six fumbles last season.
Coach Tom Coughlin said he has discussed the turnovers with Bradshaw, a former seventh-round draft pick.
"Ball security, ball security, ball security, even at the expense of trying to make the extra effort," Coughlin said after practice ended Monday night. "(You) have to know where the ball is at all times and that people are coming after you. That's the nature of our game."
The Giants also signed quarterback David Carr and tight end Ben Patrick and waived-injured rookie running back Martell Mallard. Safety Chad Jones, the third-round draft choice in 2010 who was seriously injured in an automobile accident last year was waived/non-football injury.
Coughlin said the signing of Patrick will not influence whether the Giants re-sign Kevin Boss, who has been New York's starting tight end for the past three seasons.
Patrick has been used mostly as a blocking tight end during his first four seasons with the Arizona Cardinals.
"He worked out very well, caught the ball, moved well. He's in very good shape," Coughlin said of Patrick.
Carr spent the 2008 and '09 seasons with the Giants as the backup to Eli Manning before signing with San Francisco 49ers last year.
Coughlin said the Giants wanted four quarterbacks in camp and Carr was available and knew the system. Veteran Sage Rosenfels and free agent Ryan Perrilloux are New York's other quarterbacks.
Rosenfels, who was Manning's backup last season, said the decision to bring in Carr was not a surprise. He also said he has not been asked to restructure his contract at this point, although the Giants could use Carr as leverage to get him to take a pay cut.
"I only can control so much," Rosenfels said. "What I can control is my play and how I can get the other 10 guys around me to play good football. That's my concern right now.
Jones was at practice on Monday night and he said the leg he hurt in the crash is roughly 70 percent healed.
"It was really nice to see him," Coughlin said. "I looked over and saw this big guy on the sideline and said, 'Is that Chad?' He looks great, he sounds good."
Center Adam Koets has been placed on the reserve/physically unable to perform list with knee tendinitis. He tore his ACL in a game against Seattle on Nov. 7.