Calciomercato.com says Liverpool have now jumped to the front of the queue for the £31m-rated central midfielder, although a number of other Serie A clubs, including Inter Milan, AC Milan and Roma are also tracking his progress.
Barella had been expected to leave Cagliari this summer, but instead pledged his future to his boyhood club.
"The transfer market? I feel at home with the team of my city," he said earlier this summer.
"Together with the president, I decided to prolong my relationship with them, rejecting the offers that came my way.
"I didn't even take them into consideration. My first choice has always been Cagliari."