The Switzerland international agreed to join the blue and black half of Milan on loan until the end of the season on Thursday, with a permanent €15 million deal expected to go through at the end of the campaign.
His future has been the subject of intense speculation for much of the last 12 months, with Liverpool strongly credited with an interest throughout, but it appears the Swiss long had his future mapped out.
Asked why Bayern had turned down a larger €20m bid from the Reds, Rummenigge said: "I don't owe explanations to anyone.
"But if you really want to know, it's very simple: the boy only wanted to join Inter and it was the same for us. End of story. I also refused offers from half the continent back in the day so I could join Inter. Inter are Inter."
Shaqiri, 23, joined Bayern from Basel in 2012 with a reputation as one of Europe's finest attacking prospects, but found opportunities limited at the Allianz Arena.
He departs with two Bundesliga crowns and a Champions League title on his CV, but without ever nailing down a regular starting slot as a result of Pep Guardiola's preference to field Arjen Robben and Franck Ribery in the two wide roles.
Inter boss Roberto Mancini could hand him his first taste of Serie A action in the clash with Genoa at San Siro on Sunday.