Sociedad announced on Tuesday that an agreement had been reached between the clubs and two days later Leeds confirmed Llorente had passed his medical and signed a four-year contract.
The 27-year-old is looking forward to working under Marcelo Bielsa and says he is ready to play against Sheffield United at Bramall Lane on Sunday if required.
He said: "For me, this step is very important. I think that I am doing the right thing. I hope to help my team-mates and to improve my football.
"I want to learn a lot off the coach. I think we can have a great season and I will give my best.
"The coach has been an important factor to be here. I think I can learn a lot from him and improve my football, so it's an honour for me to train and learn from Marcelo.
"It was a difficult few days for me with a lot of change in a short time, but I am always ready to help the team. I'm always ready to help the coach if he wants me to play right now."
The former Real Madrid centre-back joined Sociedad in 2017 following successful loan spells at Rayo Vallecano and Malaga in LaLiga.
Spain international Llorente will likely have to compete with captain Liam Cooper and new signing Robin Koch for a starting place in the centre of defence.
Leeds has three points from its opening two Premier League games, having bounced back from losing 4-3 at Liverpool to defeat Fulham by the same scoreline.