The 24-year-old met with a specialist on Tuesday who indicated he will require an operation, which sources told ESPN will take place on Thursday.
Arsenal insist Tierney did not suffer the problem while on international duty with Scotland but in a training session at London Colney last week.
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An exact timeframe on his recovery is yet to be determined but sources have told ESPN that Tierney is a major doubt for Scotland's World Cup qualification playoff against Ukraine, scheduled for June.
"Kieran felt discomfort after a training session on Thursday, March 31," a club statement said. "Further assessments and scans have confirmed that Kieran has damaged his left knee.
"A specialist consultation took place on Tuesday, and as a result of further discussions with Kieran and our medical team, it has been decided that Kieran requires surgery to his knee. This procedure will take place in London in the forthcoming days.
"Kieran's rehabilitation programme will start immediately after his surgery, at which stage a clearer understanding will be established on his recovery timescale. However, it is likely that Kieran will now miss the remainder of this season.
"Everyone at the club will now be supporting and working hard with Kieran to get him back on the pitch as soon as possible."
Tierney's absence is a significant blow to Arsenal, whose hopes of securing a Premier League top-four finish for the first time since 2016 suffered a setback with Monday's 3-0 defeat at Crystal Palace.
On his previous start back in January, against Nottingham Forest as Arsenal exited the FA Cup at the third round stage, Tavares was hauled off after just 34 minutes.
Both appearances raise serious questions about whether Tavares is ready to play a prominent part in the final nine games, with Arsenal sitting in fifth place, below Tottenham on goal difference.
Arteta has other options including midfielder Granit Xhaka switching to left-back -- as he did at Selhurst Park -- or Cedric Soares being deployed on the opposite flank but that would largely depend on right-back Takehiro Tomiyasu recovering from injury and the club confirmed the Japan international will not return to training until after Arsenal's home game against Southampton on April 16 at the earliest.
Speaking after Monday's loss to Palace, Arteta said: "He is seeing the specialist tomorrow and we will know more the extent of the injury. But the feeling that he had it wasn't really positive and what the scans showed either, but we have to wait and see what happens."