Upamecano, 22, has developed into one of the most highly prized centre-backs in Europe in recent seasons and has been tipped for a major move at the end of the season.
Bayern has been heavily linked with the France international, particularly with David Alaba expected to leave on a free transfer, most likely to Real Madrid.
Rummenigge confirmed transfer plans are being drawn up but was not prepared to state that any agreement had already been reached.
He told Bild that Upamecano was "at least at Leipzig until the end of the season", adding it was "a bit too early" to assume he would join the champion.
"We'll deal with this, of course, not only with him specifically but crucially which positions we want and where we have to do something," he said.
Bayern could also lose Jerome Boateng after this season, with the 32-year-old's contract expiring in June.
However, Rummenigge suggested he could extend his deal at the Allianz Arena after re-establishing his place in Hansi Flick's plans.
"Jerome is playing a good role again. The coach is satisfied with him," said Rummenigge.
Rummenigge also rubbished suggestions Bayern could look to sell Leroy Sane this year.
The former Manchester City winger has only started seven times in the Bundesliga in 2020-21, although he has scored four goals and provided five assists in those appearances.
"He has a five-year contract. It would be nonsensical [to sell]. We need this kind of player," Rummenigge added.
"Of course he will stay here at Bayern. And I'm convinced he'll find his way here."