The Stamford Bridge side were reportedly confident of wrapping up a deal for the Colombian international this summer after reaching an agreement with Atletico Madrid.
But the report claims Falcao could seal a move to Monaco as soon as today as the club's Russian owner Dmitry Rybolovlev looks to put together a squad capable of challenging Paris St Germain for the Ligue 1 title next season.
The deal is thought to be worth £8.5m-a-year after tax. But aside from the financial rewards, a key consideration for Falcao is believed to be his desire to one day play for Real Madrid.
Moving straight to the Bernabeu from city rivals Atletico is almost unthinkable given the history between the clubs, so Falcao will move to Monaco as a stop-gap between the two.
The report claims his Monaco contract is likely to have a release clause allowing him to leave France as early as January, or possibly next summer.
Chelsea's efforts to clinch the signing of Falcao were hampered by the complex third-party arrangements surrounding his ownership.
The player's agent, Jorge Mendes, is part of a group of businessman that retain a holding stake in Falcao, while the striker's wage demands were also thought to be a stumbling block. He was reportedly after around £250,000-a-week to move to Stamford Bridge.
Despite Roman Abramovich's willingness to meet Falcao's buyout clause, it now seems Chelsea will be beaten to the signing by the Ligue 2 side.