The veteran Uruguayan has been a peripheral figure at times for Man Utd, and he has been linked with a move to Barcelona this month.
However, Rangnick says Cavani remains committed to the Red Devils for the rest of the season following a meeting with the player player last week.
"I told him that if it was up to me I would want him to stay and I had a conversation with him (on Thursday)," Rangnick said of Cavani.
"He came to my office and we spoke for almost half an hour and he told me that he will definitely stay, he will want to stay until the end of the season.
"Not only because I told him that he should stay or has to stay, but he did it on his own behalf.
"He told me that I can rely on him to stay until the end of the season and give his very best, be a role model for the young players.
"If he should play from the start, he's available and he's happy to play.
"Even if he's not playing, he will be trying to be the best possible role model for the other players.
"This is, for me, good news because Edi is one of those players with his vast amount of experience, his mentality, his work ethic, who could really be a perfect role model for all the other players."
Man utd will welcome Aston Villa to Old Trafford on Monday, with Rangnick set to take charge of his first ever FA Cup match.
He, though, says he has loved the competition ever since he was a student in England in the early 1980s.
"I've always been a big fan, supporter and admirer of the FA Cup," the 63-year-old said.
"I remember back in 1980 or 1981 when I spent my year abroad in Brighton I watched the FA Cup final at Wembley.
"I've always been a big admirer of the atmosphere, that was obviously the old Wembley stadium.
"For me it's a very important competition. We will definitely try to play with our best possible team in that game and a strong team.
"We take the game as seriously as we can and hopefully get one round further ahead but in order to do that we have to be successful against Aston Villa.
"We know it will not be an easy game because especially since they change managers they have been doing well.
"They won four games and lost four games, three of them against the top team. It needs a top performance from us to be successful."