League sources told ESPN and NFL Media on Sunday that Stefanski, 37, will replace former Browns head coach Freddie Kitchens, who was fired on Dec. 29 after Cleveland posted a 6-10 record in his first full season at the helm.
Stefanski also was a finalist for the Browns' head-coaching vacancy last year, but Cleveland opted to promote Kitchens from his interim offensive coordinator position.
The San Francisco 49ers beat the Vikings 27-10 in the NFC Divisional Round on Saturday, clearing the way for the Browns to move forward with Stefanski.
Stefanski will become the Browns' 18th head coach in franchise history. He will be the fifth head coach hired by owners Jimmy and Dee Haslam since they purchased the team in 2012.
Stefanski, who played defensive back at Penn and graduated in 2005, has served as an assistant with the Vikings for the past 13 years. He took over as interim offensive coordinator in Minnesota last season and was promoted to offensive coordinator this year.
The Vikings ranked eighth in the NFL in scoring offense with 25.4 points per game and sixth in rushing offense (133.3 yards per game).
With the Browns expected to hire Stefanski, there are currently no remaining head-coaching vacancies in the league.