The 20-year-old has just spent a year on loan at the Scottish giants, with 11 goals being scored in 29 appearances in all competitions.
Last summer, it was agreed that Celtic could sign Edouard for a fee in the region of £9m before a deadline set by PSG, which is said to be June 16.
According to Sky Sports News, Brendan Rodgers has opted to take advantage of that clause after being left impressed by the youngster's contribution last season.
Should a deal go through, Celtic will break their club transfer record, which was set when Chris Sutton left Chelsea for Glasgow back in 2000.
PSG will also take a step towards improving their position with FFP, with UEFA still monitoring the finances of the French club.