On Monday alone, three defenders were pinpointed as potential replacements for the 22-year-old.
Celtic have already rejected a bid of £15million for the player and it is understood that the Premier League side are preparing an offer of £18million with relevant add-ons and potentially players as part of the deal also.
The Scottish champions value Tierney at £25million but Arsenal are reluctant to meet the price.
With interest from around Europe there is a good chance the player could move on from Parkhead this summer.
If the transfer was to go through who could replace him?
Greg Taylor - Kilmarnock
The 21-year-old has already replaced Kieran Tierney this summer when he was called up to Scotland national team duty with the Celtic star out through injury. He made his debut against Belgium and didn't look out of place. A different type of full-back to Tierney but there are similarities in their tenacity and defensive skills. Taylor improved hugely under Steve Clarke and is a consistent performer, rarely dropping his high stands. Both sides of the Old Firm have bee credited with an interest. He knows Scottish football and is proven, plus has a lot of potential. Would not be too costly.
Barry Douglas - Leeds United
Since leaving Dundee United after moving from amateur side Queen's Park, Douglas has carved quite the career for himself. He has won trophies at Lech Poznan and Konyaspor in Poland and Turkey respectively before helping Wolves win the Championship. Was signed by Leeds last summer for £3million and was a consistent presence before his season was interrupted then finished by injury issues. The Scotland international's attacking attributes would make him an ideal replacement, certainly domestically and is well versed on the European stage.
Max Lowe - Derby County
Wanted by both Celtic and Rangers, the left-back impressed on loan at Aberdeen, who also want him, last season. He initially signed on a six-month deal, returned to his parent club, struggled against Leeds and found himself back at Pittodrie. In the Granite City he became a fans' favourite with his adventurous displays from defence. Needs work defensively but would fit into Celtic's attacking system and would give the team drive and intent from full-back. His qualities would be suited to a wing-back position also.
Boli Bolingoli-Mbombo - Rapid Vienna
The Zaire-born Belgian has been a regular for the Austrian side since moving from Club Brugge two years ago. He's had plenty of European experience for both clubs and came up against Rangers last year when Rapid met Celtic's rivals. It is clear why the club are reported to be interested in the £3million-rated 23-year-old. He crosses and dribbles plenty from left-back and is heavily involved in play - only one player made more passes for Rapid last season than Mbombo. At an age where Celtic could profit further down the line.
Milad Mohammadi - Akhmat Grozny
The 25-year-old will become a free agent on 1 July having rejected a contract offer from his Russian side. Celtic have previously been interested in the player and according to reports are keen on the Iranian international once more. The Scottish champions would be signing an aggressive player with and without the ball. Only one player was involved in more defensive duels that Mohammadi and he was also second for the most amount of crosses. He'll go down in World Cup folklore for his attempt at a somersault throw-in during last year's tournament in Russ. Has interest from Belgium.
Erik Durm - Huddersfield Town
It is not often a 27-year-old seven-cap German international is available on a free. Durm will be available when his contract expires with relegated Huddersfield Town at the end of the month. Prior to joining the Terriers he made just 40 appearances in three seasons due to injury issues but played 30 times last campaign.
Filipe Luis - Atletico Madrid
The least likeliest of all the options but what a wonderful addition to Scottish football he would be. Out of contract at Atletico Madrid and still only 33 years of age. A genuine high quality player with more than 350 La Liga appearances, including 27 last season, plus 39 Brazil caps. Currently playing at the Copa America but who knows...