According to ESPN's Adam Schefter, McCoy is in line to start over Daniel Jones, who has ankle and hamstring injuries.
McCoy started in place of the injured Jones two weeks ago in New York's upset win at the Seattle Seahawks. Jones returned last week, but he took a beating, getting sacked six times in a loss to the Arizona Cardinals.
McCoy's start in Week 13 was just his fourth in the past five seasons, as he has served as a backup for the Giants and Washington Football Team.
While McCoy finished only 13-of-22 for 105 yards with one touchdown and one interception, he did enough to supplement a fantastic defensive performance and help push Big Blue's record to 5-7.
At the time, New York was tied with Washington for first place in the NFC East and held the tiebreaker. After last week's loss, however, the Giants are one game out of the lead, as Washington beat the San Francisco 49ers.
Sunday night's game is huge for New York since it needs to win to keep pace with Washington, which is set to face Seattle on Sunday.
If McCoy starts, he will go up against the team that selected him in the third round of the draft in 2010.
McCoy spent his first three seasons with Cleveland and went 6-15 as a starter. His most productive season remains 2011, when he completed 57.2 percent of his passes for 2,733 yards and 14 touchdowns with 11 interceptions in 13 starts.
Another former member of the Browns, Freddie Kitchens, will call plays Sunday. Kitchens was the team's offensive coordinator for the second half of 2018 and head coach in 2019, but he was fired after posting a 6-10 record.
The Giants hired Kitchens as their tight ends coach, but he will fill in for Jason Garrett as offensive coordinator after Garrett tested positive for COVID-19.
If the Browns beat their former quarterback and head coach, they will clinch their first season with 10 wins since 2007.