The defender enjoyed a successful spell on loan at the Stadium of Light in the second half of last season and boss Gus Poyet is keen to get him on board for the upcoming campaign.
Talks have been held between the Black Cats and Fiorentina over the past few weeks but so far no deal has been reached amid claims an official offer was tabled over the weekend.
It is thought Sunderland officials will now attempt to push through the transfer this week even though the former Bolton man has been training at the Viola's pre-season camp in northern Italy and Fiorentina remain open to keeping him.
Centre-back Modibo Diakite, who joined Fiorentina on loan when Alonso arrived at the start of the year, could yet be involved in a fresh agreement this summer.
Diakite is surplus to requirements under Poyet and would welcome the chance to stay in Serie A, but Sunderland may have to pay a fee in addition to the player swap, with Fiorentina putting a higher value on Alonso than Diakite.