Atletico Madrid and Napoli were heavily linked with star playmaker James, who was tipped to leave Madrid in the off-season.
But James stayed and started for Madrid for the first time since May 2017 in their 1-1 draw with Real Valladolid on August 24.
After two years away on loan at Bayern Munich and an off-season dominated by speculation, James wants to make an impact in the Spanish capital.
"I really hope that this can be my season with Real Madrid," James told El Chiringuito.
James is recovering from a calf injury sustained in action for Madrid prior to the international break.
The injury ruled James out of Colombia's friendlies against Brazil and Venezuela.
Zinedine Zidane's Madrid have only won one of their opening three matches following back-to-back draws to be four points adrift of leaders Atletico Madrid.
Madrid return to action when they welcome Levante to the Santiago Bernabeu on September 14.