The Daily Express says Spurs manager Mauricio Pochettino has indicated that the club will be quiet in the transfer market but the impending sale of Icelandic midfielder Gylfi Sigurdsson could create room for Lanzini.
River Plate president Rodolfo D'Onofrio admitted in April that he will "analyse" all offers for the player.
"We still haven't received anything from Lazio and nor from any other team," he told reporters.
"We only heard what is being said in the press. Of course, we are going to analyse an offer if it comes in.
"However, we are also going to listen to what the player wants to do."
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