Antonio Conte is desperate to splash the cash and mount another title charge this season.
The Blues boss also has Champions League football to focus on this time round.
And according to Spanish transfer outlet Don Balon, Chelsea have made a move for Llorente, 22.
They reckon the midfielder has an offer on the table from the Blues to join the Premier League champions.
Don Balon add that Llorente has other offers from La Liga clubs to quit Real this summer.
Chelsea have had a hit-and-miss start to the season. Antonio Conte wants Chelsea to add to their squad despite signing Antonio Rudiger, Willy Caballero, Tieumoue Bakayoko and Alvaro Morata.
Conte’s men were stunned at home to Burnley on the opening day but the Blues bounced back to beat Tottenham at Wembley yesterday. Conte feels more reinforcements are needed if they are to become the first team to retain the Premier League since Manchester United in 2009.
Chelsea have already spent big this summer.
The Blues have landed Alvaro Morata, Tiemoue Bakayoko, Antonio Rudiger and Willy Caballero.
The 22-year-old impressed on loan with Deportivo Alaves last term, making 32 appearances.
But, with Toni Kroos, Casemiro and Luka Modric ahead of him in the pecking order, his chances of making Zinedine Zidane’s starting line-up regularly this season are slim.
Diario Madridista add Llorente has been told by Zidane to be patient, but the youngster is seriously considering leaving the Bernabeu.
As well as Chelsea, Manchester City and Barcelona are both supposedly interested in snapping him up.
City have spent £200million already this summer but manager Pep Guardiola feels more is needed if they are to win the title and challenge for the Champions League.
However, the Blues’ need is greater and, should Conte demand a transfer, then Chelsea chief Roman Abramovich will be under pressure to get a deal done.