Barcelona are in the market for a left-back in January to provide competition for Jordi Alba and have centred in on Tagliafico, who has a contract in Amsterdam until 2023, as well as another forward.
Sources have said Ajax coach Erik ten Hag is reluctant to let the player leave this month with his side just two points clear of PSV Eindhoven in the Eredivisie and a Champions League last 16 tie against Benfica around the corner.
However, Tagliafico, who has lost his place in the team to Daley Blind, has communicated his desire to embark on a new challenge to the Dutch club.
Barca are looking for a short-term option at left-back until the summer and would prefer a loan with an option to make the deal permanent.
They will then re-assess their options at the end of the season, with Valencia's Jose Luis Gaya a long-term target. Like Tagliafico, Gaya is into the final 18 months of his contract, although reports in Valencia suggest he is considering renewing his terms.
With one week to go in the transfer window, Barcelona are also keen on strengthening their attack, with Alvaro Morata still an option, but are currently prioritising signing another defender.
Alba, who is in his 10th season at Camp Nou, has been heavily criticised for his performances in recent weeks, which prompted him to speak out after Sunday's 1-0 win at Alaves.
"I've felt singled out for many years now, but I am used to it and I accept the criticism," Alba told ESPN.
"If I play well, no one talks about me. If I have a normal game, I'm criticised for not setting up two goals. If I play badly, I am killed."
Barca have so far added Dani Alves and Ferran Torres to their squad in January. Any more signings will be dependent on exits, too, as they fight to remain within their LaLiga-imposed spending cap for the season.