Celtic have signed Odsonne Edouard on a four-year deal, with the club saying it is the “biggest transfer” in their history.
BBC Scotland believes Paris St-Germain were hoping to receive a fee in excess of £8m for the French forward, 20.
Edouard scored 11 goals in 29 appearances on loan from PSG last season as he helped Celtic win the domestic Treble.
“I was desperate to make my move permanent,” Edouard said.
“From the moment I walked into Celtic, the club has felt like home to me. I have been made to feel so welcome by everyone at the club and our fans have given me such brilliant support.”
Rodgers added on Celtic’s website: “Odsonne is one of the best young players I have worked with and one of the best in Europe.
“Our fans saw exactly what he is capable of last season and we look forward to working with him again as we look to build on the tremendous success we achieved last year. I am sure he will be a big player again for us this season.”
Celtic’s previous record transfer fee was £6m for Chris Sutton and John Hartson.
Ligue 1 champions PSG need to sell £52.9m-worth of players by the end of June, to comply with financial fair play (FFP) rules.
Uefa introduced ‘break-even’ rules in 2013, requiring clubs to balance their spending with their revenue.
Rodgers’ side will begin season 2018-19 with Champions League qualifiers next month and open their Scottish Premiership title defence at home to promoted Livingston on 4 August.