The Colombian international has also attracted interest from Manchester City, Arsenal and Bayern Munich, but Liverpool appears to be his favourite destination.
The 24-year-old, currently in Colombia preparing for next month’s Copa America, told La FM: “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 [with Udinese] but the time has come for me to make the jump to a big team.”
Zapata has two years left on his contract with the Serie A outfit.
Kenny Dalglish last week made Jordan Henderson his first signing as permanent manager.