A formal announcement is expected later Tuesday.
Shula, 52, spent the last five seasons as the Carolina Panthers' offensive coordinator. He was fired in January after Carolina finished the 2017 season ranked 28th in passing and fourth in rushing.
Giants head coach Pat Shurmur, who was hired in January following a stint as the Vikings offensive coordinator, angled to interview Minnesota quarterbacks coach Kevin Stefanski to be his right-hand man.
However, that request was denied by the Vikes, and Shurmur was forced to focus his search elsewhere, according to ESPN's Adam Schefter.
Shula will now have to adjust to a new role, with Shurmur set to call plays for the Giants in his first season at the helm.
"He's an adult. He's mature. He's got wisdom. He's very even-keeled," general manager Dave Gettleman said of Shurmur, according to Dan Salomone of the team's official website. "It really pays off. I've watched him on the sideline. He doesn't get shook. He doesn't get rattled. This is a job for a grownup."
Shurmur and Shula will be tasked with vaulting the Giants back to respectability after they finished last season ranked 31st in scoring (15.4 points per game) and 22nd in total offense (314.3 yards per game).