Well, this summer could finally end those rumors, with Jose Mourinho apparently admitting his interest in the Colombian in an interview with French TV station Canal+.
As the Daily Mail reported on the back of that interview in February, Mourinho said of Falcao: "I have a team, but no striker. Falcao does not have a team. A player like him can't play in front of 3,000 people. Monaco is a club to end [your career with]."
It's not often Mourinho is so candid about his transfer targets, so with him publicly expressing interest, we can expect the Blues to up the ante come the end of the season.
But what would Falcao give Chelsea that they don't already have? We take a look at six reasons why the Monaco front man would be the ideal fit at Stamford Bridge.