"I can confirm that we have with both parties, Juventus and the player, reached an agreement. What is missing is the medical and the signature of the contract," Rummenigge told reporters during the team's trip to China.
Vidal, no stranger to the Bundesliga having played for Bayer Leverkusen for four seasons until 2011, has won four straight Italian league titles and helped Juventus reach the Champions League final last season, where they lost to Barcelona.
The 28-year-old, who also helped Chile win the Copa America title for the first time on home soil earlier this month, could join up with his Bayern team mates as early as next week, Rummenigge said.
"If everything goes according to plan, he can start training next week," he added, without giving details of a fee or length of contract for the versatile player.
Vidal is set to become Bayern's second major signing of the transfer window after Brazilian attacking midfielder Douglas Costa joined the club at the start of July.
The Bavarians, who recently sold club legend Bastian Schweinsteiger to Manchester United, may not stop at Vidal and could still add to the squad this pre-season.
"It is still too early to close the door and say that was it. We have to wait and take smart decisions. We are charged with setting up a team that can be successful and play attractive football," Rummenigge added.
He did, however, dismiss reports claiming the club was considering letting attacking midfielder Thomas Mueller join Schweinsteiger at Old Trafford.
"There are some players who just do not have a price tag. We would have taken leave of our senses if we gave up such a player. There is nothing true in this (reports)," he said.
"Bastian Schweinsteiger was named football god by the fans. Maybe Thomas Mueller takes over this name now."
(Reporting by Karolos Grohmann; Editing by John O'Brien)
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