The Colombia international is rated as one of the best defenders in Serie A and he played a major role in helping them into the Champions League next season.
He is currently back in South America preparing for the Copa America, but he has revealed that he is in talks about leaving Italy.
The 24-year-old has stated that Liverpool and Bayern Munich have both made moves for his signature, but he admits that he is close to sealing a switch to Anfield.
Asked about his future by La FM, he said: "I do not know what to say, I can say that I am very close to Liverpool.
"At the end of the Copa America, we will see what happens with my future.
"I have gained experience, but the time has come for me to make the jump to a big team."
Zapata, who has been with Udinese for four years, plays in central defence but is also capable of playing in either full-back position.
A previous target of Arsenal, Zapata currently has two years left on his contract at the Stade Friuli.