Cofield's agent, Mike McCartney, confirmed Cofield's agreement via his Twitter account.
Cofield, 31, played his first five NFL seasons for the Giants from 2006-2010, helping them win the Super Bowl XLII title following the 2007 season. He signed as a free agent with Washington following the 2010 season and had played for them for the past four years before he was released in the offseason. In March, he underwent surgery to repair a torn labrum in his hip.
Meriweather started the first 11 games of the season for them after the Giants signed him off the street during training camp.
They're also thin at safety, but sources say the team was happy with the way safety Cooper Taylor played in Monday's victory in Miami. The Giants will move forward the rest of this season with a three-man safety rotation of Landon Collins, Taylor and veteran Craig Dahl, though they could also add another safety for depth.
Meriweather, 31, had two interceptions for the Giants this year. He had previously played with New England, Chicago and Washington during his first eight seasons in the NFL.
ESPN's Jeremy Fowler contributed to this report.