Bayern striker Klose out for eight weeks
Bayern had originally said Klose would need at least two weeks to recover from surgery on his right ankle.
"He is going to have an operation this afternoon and what does that mean? Well, that he will be out for six to eight weeks," Hoeness told the Sueddeutsche newspaper.
The club said later on Monday that the surgery had been a success.
The draw for the Champions League quarter-finals takes place on Friday, with the matches set for April 7-8 and 14-15.
The loss of Klose will also hit Bayern's Bundesliga campaign. The champions are currently second, four points behind leaders Hertha Berlin with 10 matches to go.
Klose suffered the injury after stepping into a small hole on the pitch during Bayern's 3-0 win at Bochum.
He will also miss Germany's upcoming World Cup qualifiers against Liechtenstein and Wales.