According to Sky, the 22-year-old is already travelling to Glasgow to finalise the switch, bringing an end to weeks of on-off negotiations between Celtic and SC Paderborn.
Baur had long since agreed personal terms with the Scottish champions on a contract running to 2031, with the deal previously stalling over a relatively small gap between the two clubs' valuations before Paderborn's asking price was ultimately met.
SC Paderborn stand to bank a club-record fee for the sale, with the total package including bonuses reaching around €8 million.
That represents a significant return on a player who has spent his entire senior career at the newly promoted Bundesliga side, and it comes after Baur was also strongly courted by Schalke, who were unable to match Celtic's financial terms despite their own interest in the midfielder.
Baur contributed four goals and seven assists as Paderborn secured promotion last season, form that established him as one of the more sought-after prospects to emerge from German football's second tier and helped drive the price up during negotiations. Paderborn had briefly indicated a willingness to keep him for their return to the Bundesliga even at the risk of eventually losing him for nothing on a free transfer, a stance that only shifted once Celtic's offer reached a level the club considered acceptable.
The move gives Martin O'Neill fresh options in a Celtic midfield facing turnover elsewhere, with Arne Engels expected to leave the club and further departures possible before the transfer window closes.
Baur's arrival is expected to be followed by further incomings as Celtic look to rebuild depth across the middle of the park ahead of a season that will include both Premiership and European commitments. For Paderborn, the sale allows the club to reinvest in their own squad as they prepare for the unfamiliar demands of top-flight football, having already lost several fringe players without generating fees earlier in the window.