French sports daily L'Equipe claims that the Principality club had been eager to sign the 21-year-old during the transfer window - and the former Paris Saint-Germain youngster was open to the switch.
Edouard - whose goal against Rangers on Sunday takes his Hoops tally to 40 goals in 92 games - established himself as Celtic's first-choice centre-forward following the departure of countryman Moussa Dembele to Lyon, but came very close to leaving Celtic himself.
L'Equipe reports that Monaco stepped up talks with Edouard in June, with serious efforts being made by the club to secure the striker's services ahead of the 2019/20 season.
A contract was reportedly agreed in principle as negotiations progressed but, out of the blue, Monaco halted discussions with no explanation.
Edouard's representatives were expecting to hear back from the club, but were forced to wait until the end of the transfer window before the eight-time Ligue 1 winners got back in touch.
In good news for Celtic, L'Equipe reports the link as now being "completely dead".
Monaco ended up signing Wissam Ben Yedder from Sevilla while bringing in Jean-Kevin Augustin from RB Leipzig and Leicester's Islam Slimani on loan to bolster their attacking options.