Rodriguez on Tuesday signed a six-year contract with European champions Real after Monaco accepted a reported £63million transfer fee that, if accurate, would make the forward the fourth most expensive player of all time.
And although the former Atletico Madrid striker has not openly confessed to wanting a switch to Real, Rodriguez feels he would be a "great" addition to Carlo Ancelotti's side.
Speaking at his introductory press conference, Rodriguez said: "He has not said anything to me and it's not really for me to talk about, but Falcao is a great goalscorer and could play here.
"It is up to the club."
Falcao missed the World Cup after suffering knee ligament damage in January and is steadily working his way back to full fitness.
Whether or not his countryman follows him to Spain, Rodriguez now has a "dream" opportunity to work alongside Bale and Ronaldo in the pursuit of further glory following Madrid's recent Champions League triumph.
He said: "I just want to help make history and bring joy to the Madrid fans.
"This is a club which is used to winning and I'm prepared mentally and physically to do so. It is a pleasure to be among so many stars and I'm sure I'll learn a lot from everyone."
Rodriguez was one of the revelations of the World Cup, scoring in each of Colombia's games as they reached the quarter-finals of the competition for the very first time, eventually losing 2-1 to hosts Brazil.
His stunning 25-yard volley against Uruguay was voted goal of the tournament on FIFA.com.
After signing his contract alongside Real president Florentino Perez and being formally presented in the Bernabeu's presidential box on Tuesday, Rodriguez retreated inside to change into the famous white kit for the first time.
He emerged to a rapturous reception from a 45,000-strong crowd, around a third of whom were wearing Colombia shirts.
Then, while he kicked balls into the crowd - as is traditional for new high-profile arrivals to do - he was rushed by several enthusiastic supporters and made a point of embracing them before they were ushered away by security.
Tweet: Rodriguez 'selfie'
Rodriguez added: "I've always followed Real Madrid and always dreamed of playing here.
"I've suffered a lot to get here and when you do that then it tastes so much better.
"I will never forget this day. I hope to work hard, to train well, and to experience a lot of joy here.
"I know I am under a lot of pressure, but I am happy to face it."
Rodriguez began his professional career with Colombian side Envigado before moving down to Argentinian club Banfield aged 17.
He joined Porto for £4million in 2010, and three years later he signed for Monaco for £38.5million after helping the Portuguese side win a hat-trick of league titles and the 2011 Europa League.
The left-footed forward scored 10 goals in 38 matches in all competitions for Monaco last term as they finished second in Ligue 1 behind Paris St Germain.
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