The Gunners have reportedly had an initial £15m bid for the Scotland international rejected by the Hoops, who insist the price starts at £25m.
Even offering the likes of Carl Jenkinson, Jordi Osei-Tutu, Emile Smith Rowe or Eddie Nketiah could not tempt the Scottish champions.
Celtic though are thought to be open to a deal – if the price for the 22-year-old is right, according to The Sun.
The report suggests that the Premier League side will come back with an improved offer in the coming days as they look to secure their man.
Tierney is a hero at Parkhead and feels a huge loyalty to his boyhood club, but may feel that the time is right to move south of the border.
The 22-year-old has been linked with a move to the Premier League for some time but has always been happy to remain where he is.
Since breaking into the first team in the 2014/15 season, he has won four consecutive Scottish Premiership titles.
Tierney has a contract at the club that runs until 2023 and is now concentrating on recovering from a double hernia operation that he had at the end of May.
Arsenal boss Unai Emery is in the market for a new left-back with Nacho Monreal’s contract up this summer.
Indeed, he is on the hunt for reinforcements for most positions in defence after it was at the back that cost the team last season.
But after missing out on Champions League football for next season, the Spaniard is having to do business on a much tighter budget than he would have liked.
Another season in the Europa League to come has left the former Sevilla and Paris Saint-Germain boss with a rumoured £40m to spend this summer.
That is unless he can sell on some of the players that he has deemed surplus to requirements.
At least the wage bill has been somewhat downsized after the departures of Aaron Ramsey, Denis Suarez, Danny Welbeck, Petr Cech, David Ospina and Stephan Lichtsteiner so far this summer.
German playmaker Mesut Ozil remains the highest earner though with his incredible £350,000-a-week deal.
Arsenal’s annual wage bill of £200m is still below Chelsea, Manchester City, Liverpool and Manchester United.