According to The Express, Madrid president Florentino Perez is struggling to find suitors for the Colombian international and could drop his valuation if there are no teams willing to match it.
Although United and Paris Saint-Germain remain interested in Rodriguez, they are unwilling to meet Madrid's reported valuation of €75m (£65m, $85.6m) for the 25-year-old, with the La Liga champions looking to recoup the amount they paid Monaco in 2014.
United may benefit from the lack of suitors as they could sign the Madrid outcast for a cheaper fee with Perez likely to drop the asking price at the last minute.
The report also claims that Rodriguez is only interested in a move to the Europa League champions as the former FC Porto star finds himself surplus to requirements at the Bernabeu.
With manager Jose Mourinho also looking to sign Alvaro Morata, Madrid would prefer not to sell Rodriguez along with the Spaniard.
However, with the two teams set to face each other in the European Super Cup on 8 August, Los Blancos could avoid playing two of their former players if Rodriguez is sold for a cheaper fee late in the transfer window.
A report from Spanish publication Diario Gol claims that while Perez would be fine with Rodriguez staying at the club for another season if no bids were to arrive, Zidane is against the idea and would prefer to offload him.
According to Italian publication Mediaset Premium, Rodriguez, who only started 13 league games last season, has already agreed personal terms with United over a proposed move to Old Trafford.
Only a transfer fee between the two clubs that needs to be agreed for the move to go ahead.