Rodriguez cost Real £56m when he joined the Bernabeu club from Monaco after a stellar 2014 World Cup campaign.
The Colombia international has been linked with United in the past and the Daily Mirror claim bosses at Old Trafford have been told they can make a deal this summer.
But United will have to find a huge chunk of manager Jose Mourinho’s transfer budget to complete the move.
According to the report, Rodriguez’s house is up for sale as he seeks regular playing opportunities elsewhere.
Mourinho has already signed Villarreal defender Eric Bailly in a £30m swoop and is expected to snap up Zlatan Ibrahimovic at the start of next month.
United are also thought to be considering a bid for Gareth Bale, with Spanish newspaper AS claiming the UK’s decision to leave the EU could play a factor in the move