The 26-year-old Colombian international is expected to leave Atletico Madrid at the end of the season, with a host of Europe's leading clubs all vying for his signature.
Blues boss Roberto di Matteo has been credited with an interest in bringing the prolific forward to Stamford Bridge in January, but may have to sell Fernando Torres in order to make room on the club's current wage bill.
Falcao's agent, Jorge Mendes is believed to have held preliminary talks with a number of parties, including Chelsea, prior to greeting Mancini when he flew out to Madrid earlier this week.
Mendes also represents Real Madrid coach Jose Mourinho, which has made the reigning La Liga champions favourite to sign the striker, though City are pushing hard to outmanoeuvre their rivals.
Mancini would also have to sell if Falcao was signed, due to his excessive wage demands (he reportedly wants a £5million salary after tax), with the Etihad Stadium futures of Mario Balotelli and Edin Dzeko most in doubt.