The Daily Mail says the La Liga side had offered to take him on loan for the rest of the season and pay his wages, but were turned down by Liverpool technical director Michael Edwards.
The Reds will reportedly only sanction Sturridge's exit if it is a permanent transfer, with the Daily Telegraph claiming that the England striker is valued by the Merseyside club at £30m ($41m).
Italian giants Inter Milan have also expressed interest in signing Sturridge, although the £30m asking price is bound to be a stumbling block in negotiations.
Sturridge has only started five Premier League games for the Reds this season — despite being fit for the majority of the campaign — and wants to leave Anfield in order to boost his chances of playing for England at the World Cup.
The 28-year-old has 18 months left on his Liverpool contract, which is potentially worth as much as £150,000 per week.
Earlier reports suggested that Reds boss Jurgen Klopp was reluctant to lose both Sturridge and Danny Ings, who has been linked with Newcastle, this month and leave his squad short of cover in the attacking positions.
Sturridge suffered a fresh muscular injury in late December 2017, but Klopp played down its significance.
"Daniel Sturridge was ill and now [has] a little muscle thing. It's not a big deal, to be honest, but in this period all these things keep you away from games," the German told Liverpool's official website.
Former Reds defender Phil Thompson urged Sturridge to leave Liverpool in the January window, saying he would not make the World Cup squad if he stayed at Anfield.
"As a Liverpool fan, I want to see our squad remain together because it isn't that big as it is and he does enhance it. However, on a personal side, having been a player, I'd be worried his chance is passing him by," Thompson told Sky Sports in November.