New York fired Ben McAdoo in 2017 after a 2-10 start, allowing Spagnuolo to take over on an interim basis.
Shurmur has been Minnesota's offensive coordinator for parts of the past two seasons and served as its tight ends coach in 2016.
The 13-3 Vikings had a surprisingly strong offense, as they ranked 10th in scoring (23.9 points per game), 11th in total yardage (356.9 yards per game) and seventh in rushing (122.3 yards per game) in 2017.
The fact promising rookie running back Dalvin Cook (torn ACL) and starting quarterback Sam Bradford (knee) missed the vast majority of the season due to injury made the team's performance more impressive.
Latavius Murray and Jerick McKinnon thrived in the backfield, while journeyman quarterback Case Keenum had a career year with 3,547 yards, 22 touchdowns and seven interceptions.
He went 9-23 over two seasons as the Cleveland Browns' head man in 2011 and 2012 before going 1-0 as Philly's interim head coach at the end of 2015.
While Shurmur struggled in Cleveland, he is far from the only head coach to stumble in that situation.
The Giants are coming off a disastrous year after finishing No. 21 in total offense, but with Shurmur's teams finishing 13th or better in points in four of the past five seasons, he could cure many of New York's ills.