Sources: Ahmad Bradshaw stays

EAST RUTHERFORD, N.J. -- The New York Giants addressed their top priority in free agency when Ahmad Bradshaw agreed to terms to re-sign with the team on Monday, according to sources.

Terms of the contract have not been not disclosed, but a source said that Bradshaw will receive a four-year deal worth $18 million, with $9 million guaranteed.

The Giants have been making moves to free up salary cap space to re-sign their starting running back. They released veterans like center Shaun O'Hara, guard Rich Seubert, left tackle Shawn Andrews, defensive tackle Rocky Bernard and restructured the contract of Brandon Jacobs to get under the salary cap.

They still have to re-sign wide receiver Steve Smith and tight end Kevin Boss. And they have to figure out what to do with Osi Umenyiora, who has been given permission to seek a trade, a league source told ESPN's Sal Paolantonio. But their biggest free agent is coming back.

"I'd rather stay," Bradshaw said on ESPN's "NFL Live" last week about New York. "I've set my foundation there in New York and I love just the excitement there. The fans are great. The community is great out there. I love New York the most but the price is going to talk."

Bradshaw was excited to see running back DeAngelo Williams agree last week to a monster five-year, $43 million deal with the Panthers, which reportedly includes $21 million guaranteed.

But the market appeared to go soft for Bradshaw. Early in free agency, he mentioned the Giants and the Dolphins as the leading candidates to land him, but Miami traded for Reggie Bush last week. Cincinnati and Washington were two other teams that reportedly had interest in Bradshaw but they settled their running back situations on Sunday by making moves. The Bengals agreed to terms with Cedric Benson and Washington traded with Arizona for Tim Hightower, according to ESPN NFL insider Adam Schefter.

All of this is good news for the Giants, who have been rather quiet on the free agency front compared to other teams, like division-rival Philadelphia.

Their biggest moves have been re-signing Bradshaw and defensive end Mathias Kiwanuka and agreeing to terms with free-agent center David Baas.

Bradshaw, 25, gained 1,235 yards and had eight touchdowns in his first season as the primary starter last year. He did lose six fumbles but managed to fix that problem toward the end of the 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."

Coming into 2010 training camp, no one really expected Bradshaw to be the Giants' starter. Brandon Jacobs had been the No. 1 running back since Tiki Barber retired after the 2006 season and there was no reason to expect a change.

With Bradshaw back, the Giants will keep their backfield intact and continue to go with Bradshaw and Jacobs, who could see a few more carries this season, according to Coughlin.

If there is a concern about the 5-foot-9 Bradshaw, it's how long his feet and ankles hold out. He went into last season with 253 carries since being drafted in the seventh round out of Marshall in 2007. He carried the ball 276 times last season.

Bradshaw had surgery on his left ankle in February. He underwent the same surgery to his right ankle in addition to surgeries on both of his feet in 2010.

Meanwhile, 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 Boss.

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.

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.

Ohm Youngmisuk covers the Giants for ESPNNewYork.com. Information from The Associated Press was used in this report.