After a poor La Liga campaign, the Spanish giants are allegedly preparing to make significant changes during the summer, with both Tottenham Hotspur's Harry Kane and Bayern Munich's Robert Lewandowski among the names linked with a switch to the Bernabeu.
According to Sport, Rashford is another player on Zinedine Zidane's radar, and Real are said to be considering a summer approach for the England international.
Rashford is regarded as an important member of Jose Mourinho's squad, but the 20-year-old has struggled for regular starts since the beginning of 2018, especially in the Premier League.
In total this season, Rashford has scored 12 goals in 22 starts and 19 substitute outings in all competitions, although just three of those goals have come since December 10.
Rashford currently has just over two years remaining on his contract, but United have the option of extending his deal by a further year.