Chelsea are set to axe Conte at the end of the season, with Real Madrid potentially his next destination.
The Italian landed the Blues the Premier League title last season but his relationship with the Chelsea hierarchy has crumbled in the time since.
Rodriguez, meanwhile, is shining at Bayern Munich after leaving Real Madrid on loan back in the summer.
And Spanish website Don Balon are sensationally reporting that Conte has contacted the Colombia international.
The Chelsea manager is supposedly preparing himself for life at Real next season.
Los Blancos have had Zinedine Zidane in the position ever since January 2016.
But, despite winning the Champions League twice in as many years, Real will axe him should they fail to scoop Europe’s biggest prize once again.
Rodriguez left Madrid for Bayern after failing to prove his worth to Zidane.
Yet Conte, Don Balon say, rates him highly.
He has told Rodriguez that he wants him at Real if he replaces Zidane at the end of the campaign.
But whether Real go for him is something only time will tell.
Conte is not the only manager in the frame to take over the job managing Cristiano Ronaldo, Gareth Bale and Co.
Germany manager Joachim Low is understood to be one of the names on Florentino Perez’s radar.
And Liverpool’s Jurgen Klopp has also been linked with the job.