The Cardinals have been in search of a new head coach since Bruce Arians announced his retirement on New Year's Day.
Shurmur's last stint as a head coach spanned the 2011 and 2012 seasons when he led the Cleveland Browns to a record of 9-23. He also coached one game for the Philadelphia Eagles on an interim basis at the end of the 2015 season following Chip Kelly's ouster.
The 52-year-old has spent the last two seasons as the Vikings offensive coordinator, with the 2017 campaign representing a high point.
While the Vikings encountered turbulence under center after Sam Bradford went down with a knee injury, Shurmur kept the offense humming with Case Keenum under center.
All told, Minnesota finished the year ranked 10th in scoring (23.9 points per game), 11th in passing offense and seventh in rushing offense.
Should Arizona strike out on Shurmur, Fouhy reported Carolina Panthers defensive coordinator Steve Wilks "is right behind Shurmur at the top of the Cardinals' list."