Arsene Wenger is interested in bringing the Albanian to the Premier League.
The 22-year-old - who is part of Albania's squad for the European Championships - is being chased by the Catalan giants, as well as Manchester United and Chelsea, the Star claims, but the Gunners are hoping that the Champions League football that they can offer will put them ahead of their two top flight rivals.
The Star reports however, that the interest in Hysaj is somewhat based around the fact that Arsene Wenger is preparing for the possibility of losing Hector Bellerin.
The Spaniard came through the youth ranks at the Nou Camp and, like Hysaj, is being lined up as a potential replacement for Dani Alves who has left for Juventus already this summer, the Star adds.
The former Empoli man would appear to be a good replacement for Bellerin, should a move back to Barca happen at some stage.
Like the 21-year-old, Hysaj is a full-back that likes to offer support in attack but is rarely guilty of shirking his defensive responsibility.
And with the defender - who played the full 90 minutes of his country's opening game at Euro 2016 - becoming Albania's youngest ever captain during a warm-up fixture ahead of the tournament, he may be able to bring some extra leadership into the Arsenal side if he was to join.