Falcao, who is yet to feature for Monaco this season because of an ankle injury, has been linked to Galatasaray throughout the transfer window.
The 33-year-old is now poised to swap Monaco for Istanbul after Galatasaray announced it has started negotiations over the transfer of the Colombian.
"Negotiations have started with the footballer and club AS Monaco on the transfer of Radamel Falcao to our club," Galatasaray's statement read.
Falcao, who is contracted until 2020, has called Monaco home since arriving from LaLiga outfit Atletico Madrid in 2013.
The Colombia international helped Monaco to Ligue 1 glory in 2016-2017, while he has scored more than 80 goals across all competitions for the French club.
Falcao, who spent time away on loan at Manchester United and Chelsea in 2014-2015 and 2015-2016, scored 16 goals for Monaco last season.