A report in The Mirror states Moyes is looking to sign the midfielder on loan after Madrid boss Jose Mourinho indicated he wanted Canales to get some first-team experience elsewhere.
The deal could suit the Everton boss, who is operating under a tight transfer budget and doesn’t want to sell any key players to make funds available.
Canales moved to the Bernabeu last summer from Racing Santander but has struggled to make a real impression in a team filled with superstars.
La Liga side Villarreal were reported to be interested in a permanent move for the Spanish Under-21 playmaker, but Madrid are reluctant to sell the teenager to another Spanish team and would prefer a loan exit rather than a full sale.
Arsenal were also looking at the 20-year-old last year but withdrew from any deal as Madrid wanted Cesc Fabregas to go in the other direction, while German side Wolfsburg are also keen to move for the Madrid midfielder.
While he is looking to add to his squad, Moyes is also on the defensive with reports suggesting he could be set to lose Spanish midfielder Mikel Arteta, with Valencia lining up a move for the Everton star.