In his strongest comments to date, Murray on Friday said Kane needed to behave better whether or not he's cleared of harassment complaints filed against him by two women following an alleged altercation at a Buffalo nightclub last week.
"He's going to have to pick and choose his spots when he goes out a lot better than he does, and he's going to have to behave himself a lot better than he has obviously," Murray said.
"Whether he has done these things or not, or he is guilty of these things or not, it's not something I like getting up in the morning and reading about, that's for sure."
According to a police report, Kane allegedly grabbed one woman by the throat. The second woman accused the 6-foot-2, 204-pound Kane of trying to force her from the bar. Police are still investigating what happened, and no charges have been filed or arrests have been made.
Kane's attorney, Paul Cambria, has said the player did nothing wrong.
This is the 24-year-old Kane's second run-in with legal trouble in six months. In March, prosecutors announced after a three-month investigation that Kane would not face a sexual assault charge stemming from a December incident because there was no evidence to support the allegation.
The Sabres have already disciplined Kane once by suspending him for one game in February after the player slept in and missed practice.
Kane spent the previous night attending the NBA All-Star Game in Toronto, which is about a two-hour drive from Buffalo. Pictures of Kane celebrating following the game surfaced on the player's Instagram account before they were deleted.
Kane apologized and accepted full responsibility for what happened.
The fourth player selected in the 2009 draft, Kane completed his first season in Buffalo after being acquired in a multiplayer trade with Winnipeg in February 2015. He has two years remaining on his contract, and scored 20 goals with 15 assists in 65 games last season.
Murray says he intends to meet with Kane to discuss his concerns once the legal process is completed.
"Certainly, we don't like the fact that this is twice now that he's been in incidents like this," Murray said. "It's not good for the organization. It's not good for him."