Hernández's 80-million euro ($90.5-million) transfer from Atletico Madrid makes him the Bundesliga most expensive player, but Rummenigge was aggrieved at criticism from Bayern goalkeeper Manuel Neuer's agent Thomas Kroth, who questioned the club's ambition in the transfer market.
Kroth told the Süddeutschen Zeitung newspaper on Saturday that the Bayern squad was not yet competitive enough "to also compete for Manuel's objectives." Rummenigge said he had phoned Kroth and established the agent wasn't "Neuer's mouthpiece on this issue."
Germany winger Leroy Sané remains a target for Bayern, which is waiting on a signal from the player on his future plans. Bayern would have to spend big to prize the 23-year-old Sané from Manchester City.
Rummenigge said the club was waiting on the first big transfer that would cause a domino effect across the market. "When it happens, a lot of transfers will follow," Rummenigge said. Apart from Hernández, Bayern has also signed his fellow World Cup winner Benjamin Pavard from Stuttgart and young German forward Fiete Arp from Hamburger SV.
The club needs to replace veteran wingers Franck Ribéry and Arjen Robben, and Rummenigge said new signings would come before the transfer window closes on Sept. 2. "We'll have to be involved in one or two transfers to broaden the squad. We have quality," Rummenigge said.
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