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Spanish daily AS says Newcastle and Deportivo La Coruna are leading the chase for the 23-year-old Llorente, who has only started two La Liga games for Real this season.
The paper claims Madrid's hierarchy are keen for Llorente to go out on loan for the second half of the season and gain valuable match experience, although the player himself is not in favour of changing clubs.
Deportivo Alaves, where the young midfielder was on loan last season, and Real Betis are also interested in signing him.
Newcastle have signed just one player this month – left-back Kenedy from Chelsea – despite manager Rafael Benitez making it clear that the club needed to invest heavily in the transfer window to remain competitive in the Premier League.
The Magpies have had two bids turned down for Feyenoord striker Nicolai Jorgensen and lost out on Daniel Sturridge to West Brom.
Benitez said this week that he remained committed to Newcastle despite his frustrations with the club's transfer policy.
"My future is very simple and very clear," the Spaniard was quoted as saying by Sky Sports. "Still we have one day for doing business. After that, still, we can analyse where we are.
"But I don't think that will change too much because I have a responsibility to the fans, to the players, to the club. I understand. I have been here for a while and I understand the feelings of everyone.
"But it's like this, so we have to be sure that if we go there, I am the first one that will push and give everything and then that will be an example to the players too.
"Sometimes you just have to say, 'Okay, I will do what I have to do'. I have the belief and confidence that even in the more difficult circumstances, still we can stay up and we try to think about that."