Rodriguez helped his country finish runners-up in the Copa América this summer and was voted player of the tournament.
He recently terminated his contract with Brazilian side Sao Paulo before joining Madrid club Rayo, who are celebrating their centenary year.
"(Rayo makes their) star signing in what is their most special season, celebrating 100 years of history, of bravery, courage and nobility," said the Spanish club in a statement.
Rodriguez was the Golden Boot winner at the 2014 World Cup, leading Real Madrid to buy him from Monaco, with the midfielder lifting two La Liga titles and two Champions Leagues with Los Blancos.
The Colombian spent two seasons on loan at Bayern Munich before departing Madrid in 2020 for Everton, also playing in Qatar and Greece before moving to Sao Paulo in 2023.