During a radio interview on Monday with SiriusXM NFL Radio, Leach said the Giants called his agents the day he was released by the Super Bowl-champion Baltimore Ravens.
The Giants' plans at fullback took a hit when Henry Hynoski suffered a knee injury this spring and needed surgery.
The Giants are not the only team interested in Leach's services.
The Miami Dolphins are considered the heavy favorite to sign Leach, a source familiar with the situation told ESPN NFL Insider Adam Schefter.
Miami has the greatest need at the position. It has two coaches, Joe Philbin and Mike Sherman, who once brought Leach to Green Bay. And the Dolphins have more money to spend than other interested teams.
A deal between the two sides is likely to get done this week.
Leach, who failed to agree on a restructured contract with Baltimore, earned Pro Bowl honors the past two seasons with the Ravens. He had 21 carries for 67 yards and a touchdown. He also had 36 catches for 212 yards.
The 31-year-old has played with the Packers, Saints and Texans.
"Everything is picking up here a little more," Leach said. "We'll see how this week goes."