The Daily Mail, as quoted by The Sun, is reporting that the Gunners are ready to compete with the likes of Atletico Madrid and Real Madrid for the signature of the defender this summer.
The same report says that Arsenal have singled out the £8m-rated left-back as a potential target this summer as Unai Emery begins to think about bolstering his squad ahead of next season.
However, the same article says that both Real Madrid and Atletico Madrid are also keeping tabs on the 26-year-old ahead of the summer transfer window.
The defender has scored one goal and made three assists in 18 Dutch league appearances for Ajax so far this season.
The same story says that Arsenal may face an uphill task in convincing Tagliafico to move to the Premier League because he has previously expressed his desire to play in La Liga.
Meanwhile, Arsenal are preparing to host Rennes at The Emirates on Thursday night as they look to overturn a 3-1 first-leg deficit and reach the Europa League quarter-finals.
The Gunners are also aiming to break back into the top four this season after missing out on Champions League qualification for the last two years running.