The 34-year-old has scored 19 goals in 49 appearances for the Red Devils since Paris St Germain’s all-time top scorer joined on a free transfer in October 2020.
Cavani was strongly tipped to leave Old Trafford at the end of last season only to agree a one-year contract extension, but talk has once grown about the striker’s future.
“We had quite a few conversations in the last couple of weeks – probably the player with whom I have spoken most,” the United boss said ahead of Monday’s clash with Wolves.
“I told him from the very first day that, for me, he is a highly important player.
“He is probably the only one who can play as a striker back to goal and face to goal.
“His professionalism, his work ethic is just amazing and I told him that I desperately want him to stay until the end of the season. He knows that.
“He also knows how highly I rate him and how highly I respect him, and that was also the reason why I played him (against Burnley) from the beginning together with Cristiano (Ronaldo).”
Asked if he envisaged letting some players leave this month, Rangnick said: “Yes, but it depends and I think you understand that I don’t want to speak about individual players now in the public.
“I know from some players that they would want to leave.
“We spoke about Edi and he knows that I will definitely not let him go. For me, he is a highly important player for the rest of the season, still playing in three competitions, so we will definitely need Edi.
“I would rather have another Edi on top of that but for me it’s clear that Edi has to stay.
“With regard to the other players, yes, our squad is maybe a little bit too big with regards to numbers.
“But we still have Covid, we have three competitions, as I said, and I think we could do with a bigger squad than maybe in times without Covid.”
Cavani looks set to stay but Anthony Martial would be allowed to leave if the right offer comes in.
The France international has expressed his desire to leave United and Rangnick has left the wantaway forward out of recent squads.
“He made it very clear that he wants to leave and, in a way, I can understand his wish to leave and to try to play more regularly somewhere else,” Rangnick said.
“But, again, it’s a question not only of what he wants to do but it’s also a question of which kind of clubs are interested in him and do they meet the demands of the club, so we have to wait and see.
“He is not injured currently so I decided to go for this group of players that were in the squad (against Burnley), so it had nothing to do with injuries.”
Dean Henderson, Jesse Lingard and Juan Mata are others that transfer talk continues to revolve around, so too injured midfielder Paul Pogba as he enters the final months of his contract.
Donny Van De Beek is another widely-discussed player having found game time hard to come by since joining from Ajax in 2020.
“Well, just work hard,” Rangnick said when asked about how the Holland international can get into his side. “So far as I have got to know him he’s got a top mentality, he works hard in every training session.
“He’s a team player through and through and there will be games where he will get his chance to play I’m sure.
“I also had a longer conversation with him last week about that and, again, of course it’s difficult for him right now because he also wants to play for the Dutch national team.
“He wants to qualify for himself and play in the World Cup in Qatar and in order to do that he needs to get game time here.
“But I still believe that we should keep him, that he should stay here, definitely until the end of this season and try to get as much game time as he possibly can.”