Los Blancos have endured a torrid start to the campaign which has seen Julen Lopetegui sacked by the Champions League holders.
Now The Sun claim Real chiefs are plotting an audacious £90m double raid for Rashford and Eriksen.
Officials from the Spanish side watched the United forward in action for England in their 2-1 win over Croatia in the Nations League on Sunday.
Real are said to see Rashford as a possible long-term replacement for Cristiano Ronaldo after his departure to Juventus.
New Los Blancos manager Santiago Solari is ready to sanction a £50m swoop for the 21-year-old.
Rashford has only made five Premier League starts for Manchester United this season with Jose Mourinho often critical of the youngster.
But it is not known whether the England international would be ready to quit Old Trafford for the Bernabeu even if an offer was accepted.
Real are also plotting a £40m offer for Tottenham midfielder Eriksen.
The Denmark international has 18 months left on his Spurs contract but talks on a new deal have stalled, according to the reports.
Eriksen has been with Tottenham since making an £11.5m transfer from Ajax back in 2013.
Solari is looking to strengthen after reportedly getting frustrated with the efforts of Gareth Bale and Karim Benzema.
Real Madrid currently sit sixth in La Liga, four points behind leaders Barcelona.
Juventus have also been linked with a raid for Rashford in recent hours.
The Times say Juve scouts have been asked to compile a report on the forward with a view to a possible transfer.
Rashford has netted just twice in 13 appearances for United in all competitions this season.