The 22-year-old arrives in a deal worth around £6.8m, joining from German club Paderborn, after helping them win promotion from the second tier of German football to the Bundesliga.
He scored four times and provided five assists across 33 league appearances last season, and was named the club's player of the year.
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Sky Sports News understand the Scottish champions will pay an initial fee of around £4.5m for Baur, with £2.3m in add-ons making up the deal.
Similar to deals struck with fellow summer signings Camilo Duran, Kasper Høgh and Haissem Hassan, a large proportion of the bonuses will be paid out if Celtic reach the league phase of the Champions League.
Celtic face LASK Linz in their play-off tie next week, and Baur - who came through the youth ranks at Freiburg before moving to Paderborn in 2024 - could now be registered for that Champions League squad.
On joining the Scottish Champions, Baur said: "It's absolutely the right move.
"It's a great club and I've heard incredible things about the fans.
"I'm in the same agency as Nicolas Kuhn so I called him to ask a couple of questions and he was so positive. He said, if you have the chance, just do it.
"I've had a good pre-season with good matches and tough training sessions, everything. So I'm in good shape and I'm ready to go."
Celtic manager, Martin O'Neill added: "I am delighted that we have been able to bring Mika to Celtic.
"He is a player with some excellent attributes, great technical ability, he will be important in linking our play and he has real versatility too.
"Having spoken to Mika, I can see the appetite he has for the challenges here at Celtic and I am sure he will be another great addition to the squad.
"Mika's club in Germany were very keen to keep him, which I fully understand, as he played such an important role in their Bundesliga promotion, but I am really pleased with the perseverance and commitment we have shown to secure his signing."
