The French international left Arsenal in the summer to sign for John Toshack's team but has failed to score in his first two months in Spanish football.
He has had injury problems, which have restricted his appearances, but Anelka claims he is being hounded by the Spanish press.
"I send a sincere greeting to the people of Lazio," the 21-year-old told the Spanish magazine Guerin Sportivo. "I want to visit them as soon as possible. I don't know whether it will be now or in the future but I must realise my dream of playing in Italy.
"Lazio is a great team full of many fabulous players, and they have a superior squad technically and mentally [to Real Madrid]. I realise I haven't scored a goal yet, and it is my duty to do so. But I don't want to speak about that. I prefer to work and develop my game and play as much as possible."
The news comes as a further blow to the under-pressure coach Toshack, who has seen his team fall 10 points behind leaders Rayo Vallecano to 11th in the Primera Liga.
Last season Anelka netted 17 Premiership goals in 35 appearances for Arsenal before spurning potential moves to Lazio and Juventus and joining Real Madrid instead.
Another former Premiership player, the Hamburg striker Tony Yeboah, wants to play in the United States. The Ghanaian, who has also played for Eintracht Frankfurt and Leeds United, told Bild newspaper that he hoped to play in Major League Soccer. The paper said his agent had already been in contact with the New York MetroStars.
"It is still my desire to play in the United States," Yeboah, 33, said. His contract with Hamburg expires at the end of this season.