With former Milan legend Franco Baresi joining Fulham as director of football, the amount of links with Italy is threatening to overwhelm The Cottagers.
Speaking to local newspaper Brescia Oggi, Giunti said: "There are too many players at Brescia, I would prefer to leave the club.
"My agent is working to find me a new team.
"I know that Fulham is chasing me. Franco Baresi is loking for some good Italian players, I am not the only one he would like to bring to Fulham."
The 31-year-old midfielder has confirmed he is ready to sign for Fulham, but he has also admitted that the two clubs are close to finding an agreement with his representatives.
However, Brescia coach Carlo Mazzone has not put too much trust in the affair and he is looking to sell Giunti to Serie A rivals Atalanta.
Mazzone has asked the Neroazzurri to swap Alessandro Rinaldi for Giunti, but Atalanta coach Giovanni Vavassori is unwilling to do so.
Mazzone has already secured the services of Hellas Verona defender Dario Dainelli, but he has asked the club to buy more players to strengthen the back-line.
