Alexis is out of contract at the end of the season and will be able to leave the Gunners on a free transfer.
Speculation has suggested the north London club could sell him during the January transfer window, though, in order to ensure he does not leave for free.
Man City aggressively pursued the Chilean during the summer but Arsenal refused to sell due to their inability to sign their preferred replacement: Monaco star Thomas Lemar.
But Rummenigge has hinted that Alexis' heart is set on a switch to the Etihad Stadium by indicating that he is no longer a Bayern target.
"I think Sanchez has already decided which club [to join], so I suppose that the door will be closed in January," he told Sky Germany.
"That's not an issue, and he is no direct substitute for Robert Lewandowski as he is not a real centre forward."
Bayern have little depth up front beyond Lewandowski, with reserve-team forward Kwasi Okyere Wriedt on the bench against Borussia Dortmund due to Thomas Muller's injury.
Rummenigge, though, is not expecting the Bavarians to make a splash in the January transfer market to bolster Jupp Heynckes' options.
"The possibilities will be limited," he added. "If you have a good player, you will not give him up - or only for a huge amount of money.
"We will not be ready to spend horrendous money on a transfer. But maybe there is a good candidate who does not cost a lot of money and makes sense."
Lewandowski scored the second goal in Bayern's 3-1 win over rivals Borussia Dortmund on Saturday.
The 72-year-old coach has enjoyed a near-perfect record since taking over from Carlo Ancelotti, winning six matches out of six in normal time and a seventh on penalties against Leipzig in the DFB-Pokal.