Rodriguez, 25, joined Los Blancos from Monaco after a stunning 2014 World Cup campaign. and enjoyed an impressive first season in Spain under Carlo Ancelotti. Since Zinedine Zidane's appointment in January 2016, however, the Colombia international has fallen out of favour, starting just four league games this term.
Isco, Marco Asensio and Mateo Kocavic are now all considered ahead of Rodriguez in Zidane's midfield pecking order. In Sunday's (18 December) Club World Cup final against Kashima Antlers, the former Porto star was an unused substitute once more, before admitting post-match a move in January is now a real possibility.
"I can't assure [you] I will stay at Real Madrid," Rodriguez said. "I have offers and seven days left to think about my future. I am happy in Madrid, but I want to play more. I have a bitter feeling as I wanted to play the final, but I am happy for the title. It is the 15th title in my career."
Spanish publication El Confidential reported in August Rodriguez's agent and United's chief executive Ed Woodward held meetings over a move for the Colombian, claiming United had agreed to make a €65m (£54.7m) offer.
Rodriguez instead chose to remain at Real and fight for his place, a move he may now regret.
His decision to speak out after Real's Club World Cup win on Sunday prompted a frustrated response from club captain Sergio Ramos, who questioned the timing of his teammate's comments.
"It is not a day for personal issues," Ramos said. "I hold him in high regard and I hope he stays. But today wasn't a day to focus on other matters."