The Scottish Sun says the Spanish side had the midfielder watched in Accies' 2-1 victory over Hearts on Sunday - and immediately set the ball rolling on a deal.
Segunda Liga Betis have already made contact with Hamilton over the move, although discussions are at an early stage.
They were blown away by McArthur's performance as he scored the opener and ran the show for Billy Reid's men.
A source close to the deal said: "Real were raving about the way James had played, they're very keen.
"They believe he's just the sort of player they could do with and will try to get him as soon as possible.
"It's a huge opportunity for James to go to Spain and make an impact."