Rodriguez and team-mate Joao Moutinho, both players linked very strongly with England in recent seasons, are said to be on the verge of joining Claudio Ranieri's side.
Monaco, who have won promotion back to Ligue 1, are reportedly ready to spend €70million on the Porto duo, and the French media claim both players were at the club's training complex in La Turbie on Thursday ahead of a deal being announced.
Moutinho said earlier this week that any switch was out of his hands, whilst Rodriguez has admitted that he is ready to go.
"Honestly, I think I'll go," the Colombia international told Ojogo.
"I am not sure of the timing, but I think I'll leave. A player always want to look ahead and play for a bigger club.
"Porto is a great club, but there are even more important (clubs) and I want to continue and to achieve things.
"If I stay I'll be happy, but if I go my heart will be with Porto."
Monaco, backed by Russian billionaire Dmitry Rybolovlev, are widely reported to have already agreed terms for Atletico Madrid star Radamel Falcao.