The Colombian international is Mourinho's top target as the Chelsea boss looks to strengthen a goal-shy forward line.
Demba Ba, Samuel Eto'o and Fernando Torres have failed to fire on all cylinders at Stamford Bridge this season, and Mourinho believes Falcao could be the man to spearhead a Premier League title challenge.
But the report claims a Chelsea bid for the 27-year-old striker would not be their first. The Stamford Bridge club have long been linked with Falcao, and have reportedly made two offers in the past.
Back in 2011, Chelsea owner Roman Abramovich reportedly sanctioned a €70m bid for Falcao when he was at Atletico Madrid.
The Russian oligarch eventually changed his mind, preferring to give £50m signing Fernando Torres a chance to take the lead role.
Then 12 months later, following Falcao's demolition of Chelsea in the Super Cup, the report claims Abramovich had an agreement with Jorge Mendes, Falcao's agent, to bring him to Stamford Bridge in the summer of 2013.
Monaco eventually signed the Colombian for £50m, but that hasn't put Chelsea off.
And with the report suggesting Falcao is unhappy in Ligue 1, Mourinho is ready to take advantage by bringing the striker to west London.