It is believed Juve are considering Dossena and international team-mate Fabio Grosso, 31, to fill the left-back spot.
"I don't know if you can call it an ultimatum, but I think that Juve will decide what to do within the next week," agent Roberto La Florio told Calciomercato.it.
"The left-back position is a necessity for the Bianconeri and the directors are not willing to drag this on. However, I still think it's a two-horse race between Andrea and Grosso.
"Dossena's age and wages are on his side. The last time I met with Liverpool we did not talk about numbers, I can only say they paid more for him last summer.
"Andrea has subtly expressed his desire to join Juve in recent days, but he also said that he loves it in Liverpool.
"We'll definitely know more about it next week and Juve remain the most likely destination for him."