Davidson leaves Columbus with four years left on his contract, and New York is expected to introduce him on Wednesday, The Athletic's Aaron Portzline reports.
Glen Sather stepped down as New York's team president last month.
Davidson became the Blue Jackets' president of hockey operations in fall 2012 after spending about six years in the same capacity with the St. Louis Blues. He hired Blue Jackets general manager Jarmo Kekalainen in February 2013.
Kekalainen will take on an alternate governor role in Davidson's absence, but won't be relinquishing his primary duties.
"I'm going to be the GM and that's going to be my role," Kekalainen said Friday, according to Brian Hedger of The Columbus Dispatch. "That's going to be my focus. That's my dream job. I'm not done."
The Blue Jackets made the playoffs in four of Davidson's seven seasons at the helm, and they swept the Tampa Bay Lightning for their first-ever series win earlier this spring.
Davidson was a color commentator on Rangers telecasts for about 20 years, and the lead analyst on FOX's NHL broadcasts alongside Mike Emrick for about five years in the '90s.
The former goaltender played eight of his 10 NHL seasons with the Rangers from 1975-76 until 1982-83, leading New York to the Stanley Cup Final in 1979 on an injured knee.