The Scotland international was injured on Monday against West Ham United in the 3-1 Premier League win at the London Stadium and will undergo an operation in the next week.
Tierney fell awkwardly in the 26th minute and was substituted shortly after with his arm in a makeshift sling.
The injury is yet another setback for the former Celtic player, who was one of Arsenal's major summer signings at a cost of just over £24 million.
Tierney tweeted his thanks to fans for their support after his latest injury:
The 22-year-old has suffered a fragmented start to his Gunners career after spending pre-season rehabilitating from hernia and hip problems.
Tierney has made eight Premier League and UEFA Europa League appearances for his new team this season since returning to action in late September.
Interim manager Freddie Ljungberg could now give Sead Kolasinac an extended run at full-back. The Bosnian was first choice in Tierney's absence earlier in the season and replaced him in the victory over West Ham.
Kolasinac could face a tough test on Sunday when Arsenal welcome Manchester City to the Emirates Stadium in the Premier League.