Shula, who is expected to coach the Giants' quarterbacks as well, was fired from the Carolina Panthers last month.
The 52-year-old Shula spent five years as the offensive coordinator of the Panthers, with new Giants general manager Dave Gettleman serving in the same role with Carolina for the first four of them before being dismissed in July.
Shula was not new head coach Pat Shurmur's first choice as the latter reportedly wanted to hire Kevin Stefanski, the Minnesota Vikings' quarterbacks coach. The Vikings denied a request by the Giants to interview Stefanski for their offensive coordinator opening, ESPN's Adam Schefter reported Saturday.
Giants quarterback Eli Manning endured a difficult season in which he saw his 210-game consecutive start streak halted. New York, which finished with a 3-13 mark, scored just 15.4 points per game - second only to the winless Cleveland Browns (14.6).
Under Shula's watch, Carolina's offense failed to find its rhythm this past season after leading the league in scoring in 2015. The Panthers finished 19th in total offense (323.7 yards per game) and 28th in passing (192.3) during the past regular season.
Shula spent seven seasons total with the Panthers, working his first two in 2011-12 as the team's quarterbacks coach.
He joined Carolina after four seasons as quarterbacks coach for the Jacksonville Jaguars from 2007-10. Shula was the head coach at the University of Alabama from 2003-06 after stops as an NFL assistant with the Tampa Bay Buccaneers (1988-90, 1996-99), Miami Dolphins (1991-92, 2000-02) and Chicago Bears (1993-95).