The French attacker has enjoyed a fine season since leaving Lyon to switch to La Liga.
And according to Estadio Deportivo, his form has alerted Los Blancos, who are sure to invest in their forward line at the end of the season.
In their last game before the coronavirus pandemic saw the Spanish season halted, Fekir helped lead Betis to a 2-1 win against Real.
His performance in particular caught the eye of Real president Florentino Perez.