But after Carlo Ancelotti told the San Siro outfit to give up on trying to land the Chilean, Rummenigge has since echoed the Bayern boss' thoughts by insisting they would turn down all bids for the star man.
"I can 100 per cent rule out Arturo's departure, for us he is a fundamental player," he told reporters. "I have not heard from anyone at Inter so far, I just read what has been written.
"The decision to continue with Vidal was taken long ago. Ancelotti talked with him: the door is 100 per cent closed.
“This is regardless of the offer: when Bayern decide not to sell a player, the fee does not matter."
Bayern have seen another midfielder linked with a move to the Italian city, as AC Milan have shown an interest in Renato Sanches, though CEO Marco Fassone said negotiations have broken down because of the asking price.
Although the Bundesliga giants are open to letting him go out on loan, he could play a part for Ancelotti's team in 2017-18.
"We are considering his situation," Rummenigge said. "He wants to play consistently but at Bayern Munich that is not easy.
"We will discuss it with Carlo, there is still time, no hurry.
"At least 10 teams would take him. He is a very in-demand player. A year ago he was at the top level, let's not forget that.
"If we were to let him go, it it would only be on loan. We are talking about a very useful player. We do not want to let go of him.
"Against Chelsea he did very well, he surprised Antonio Conte who asked Carlo about him after the game. That shows his value."
Bayern have already allowed Douglas Costa to move to Serie A, with the winger having joined Juventus on an initial loan deal, and Rummenigge believes they may regret losing him.
"He did very well during his first six months, playing great, then every now and then he would go quiet, but if Allegri manages to discipline him, he has everything needed to make a difference and become a world class player."