Shurmur, who is the Vikings offensive coordinator, has interviewed with the Arizona Cardinals, New York Giants, Chicago Bears and the Detroit Lions in the last few days. The Cardinals saw starter Carson Palmer retire on Tuesday while the Giants could be ready to move on from fellow quarterback Eli Manning.
The 52-year-old Shurmur and Keenum "both think very highly of each other," a source told ESPN.
Shurmur helped lead the Vikings to the NFL North title with a 13-3 record, doing so by getting an unprecedented performance from journeyman quarterback Keenum, who stepped in for injured Sam Bradford.
Keenum, an undrafted free agent in 2012, has completed 67.6 percent of his passes for a career high in passing yards (3,547) and touchdowns (22) while throwing just seven interceptions
That helped raise the stock of Shurmur, who has spent 30 years as a coach, including the last 19 in the NFL.
Shurmur started his NFL coaching career with the Philadelphia Eagles in 1999, working as the tight ends and offensive line coach until becoming the quarterbacks coach from 2002-08. He was the then-St. Louis Rams offensive coordinator for two seasons before his only head coaching assignment, with the Cleveland Browns in 2011 and 2012.
He was 9-23 in two seasons in Cleveland.
Shurmur went back to the Eagles as offensive coordinator from 2013-15. He joined the Vikings as tight ends coach in 2016, taking over the offensive coordinator role for the final nine games.