The 22-year-old is set to become one of the most expensive footballers of all time after the Champions League winners managed to fight off interest from Liverpool, Manchester United and Chelsea.
The Athletic report that a verbal agreement has been reached between the two parties, with just paperwork and regulatory procedures left before the transfer is officially announced.
Jurgen Klopp's Reds had been heavily linked with the emerging star as their season wound to a close, but following defeat to Real, they seemingly fell out of the race.
It then came down to PSG and Madrid, with the latter able to table a massive bid after they were snubbed by Paris' star forward Kylian Mbappe.
Tchouameni had become a first team regular at Monaco over the past two seasons since joining from Bordeaux, becoming a key cog as the principality side recorded back-to-back third place finishes in Ligue 1.
So much so that his stand-out all round play in midfield has seen him break into Didier Deschamps' France line-up despite intense competition from the likes of N'Golo Kante and Paul Pogba.
Once the deal is concluded, Tchouameni will be joining countryman Eduardo Camavinga, 19, in Carlo Ancelotti's midfield ranks, lining up a bright future as Luka Modric and Toni Kroos edge towards retirement.