Tierney is the Gunners' top priority for the position, and the club had an initial £15m rejected in June.
But while Arsenal consider returning with a increased bid, Napoli have decided that they are interested in the 22-year-old.
Carlo Ancelotti is said to be a keen admirer of Tierney and has sent scouts to Glasgow to watch a player who Celtic rate at around £25m.
The Mail also reports that Arsenal held talks this weekend with Basel defender Eray Comert, who is keen to move to the Premier League.
However, the Swiss U21 international, who is also eligible for Turkey, is not for sale, according to Basel, and it will take a huge offer for Arsenal to prise him away.
Arsenal are in a three-way bid for Dortmund's Danish forward Jacob Bruun Larsen, according to the Sun, with Manchester United and Liverpool also in the running.
The 20-year-old has impressed in the European Under-21 Championship, scoring an impressive goal in the group stage against Serbia.
Meanwhile, Aaron Ramsey has been officially unveiled as a Juventus player, having moved to Turin on a free transfer.
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