The 21-year old Hungarian gloveman spent most of the season as Pepe Reina's understudy at Anfield but according to the player's management agency, he will be on show at this summer's Toulon tournament so clubs can have a look at his talent.
"We're not surprised (at the high number of interested teams) because Peter has been considered as one of the best of his age in international football," said agency owner Tibor Pataky.
"We can't say which clubs are interested but we can say that seven teams indicated that they will travel to France in order to watch him live.
"These clubs are from England, Germany, Holland and Belgium. The most likely possibility is that Peter will spend the season at another club, but it's still a question if in England or out of England.
"Of course the final decision will be made by Liverpool. But we're not worried about this as his career has been managed very gently. They see his values and also his high potential."