Hudson-Odoi was the subject of four bids from the German champions, which were firmly rejected by Chelsea.
Bayern were desperate to lure the England Under-19 international to the Allianz Arena and even offered £35million to try to acquire his services.
They were unsuccessful in their pursuit of the pacy midfielder, who Rummenigge believes Salihamidzic was obsessed with signing.
“The only one Hasan would have liked to buy was this player from Chelsea,” Rummenigge told reporters.
“I have to say that he nearly fell in love with this player and his qualities. All the other things were rumours.
“We don’t have any demand on these positions for the second half of the season. This is a topic for the future.
“We’ll handle it in the next months and will decide.”
Salihamidzic insists that’s not the case and is adamant the only person he loves is doting wife.
“You know the only one I’m in love with is my wife,” the former Bayern Munich midfielder said.
“But I am very, very convinced of the qualities this player has.
“He has a finesse which we need to play good, successful and sometimes even spectacular football.”
Hudson-Odoi was keen on joining Bayern last month, handing a transfer request in order to force through a move to Bavaria.
Chelsea rejected his request and are bracing themselves for Bayern to renew their interest in the 18-year-old in the summer, which the Bundesliga giants have confirmed they will do.
“I guess Hasan [Salihamidzic] is a bit like Uli [Hoeneß],” Rummenigge told Goal last week.
“We also had transfers in the past that didn’t work at the first attempt but at the second.
“And that’s what we will work on now. To lay down a marker costs money.
“We want the best quality players here at Bayern and Hasan is absolutely convinced of that player.”
Hudson-Odoi and his Chelsea team-mates will be back in action next on Sunday afternoon, travelling to the Etihad to face title-chasing Manchester City.