Shurmur would reportedly accept the job if it is offered.
Chris Tomasson of the Pioneer Press reported Sunday that Shurmur and New England Patriots offensive coordinator Josh McDaniels were the two finalists, though it appears the former has moved ahead in the team's mind.
If Shurmur accepts the job, New York won't be able to announce the hire until Minnesota is eliminated from the playoffs. The Vikings are set to play the Philadelphia Eagles in the NFC Championship Game on Sunday.
Shurmur struggled as a head coach in limited chances, totaling just a 10-23 record. He spent two seasons as the Cleveland Browns head coach in 2011-12, winning only nine games. He also went 1-0 as the Philadelphia Eagles interim coach in 2015.
The 52-year-old, however, has been successful as an offensive coordinator. Despite dealing with plenty of injuries, the Vikings finished the regular season with the 10th-most points in 2017, showing balance with the No. 7 rushing attack and No. 11 passing game.
Shurmur has also worked as an offensive coordinator for the Eagles and St. Louis Rams in his 19 years in the NFL.
If he becomes the next head coach in New York, he will have to turn around an offense that finished 31st in points on its way to a disappointing 3-13 season.