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The full-back limped off during the Gunners' disappointing 3-0 reverse at home to Liverpool on Sunday (AEDT) and required further checks on the severity of the issue.
An Arsenal statement said Tierney will not require surgery on the injury but with the Premier League campaign coming to an end on 24 May, he may not feature again for Arsenal this term.
It is also unfortunate timing for Scotland, who play their first match of the rescheduled Euro 2020 finals against the Czech Republic on 15 June, with the first of two warm-up fixtures taking place against the Netherlands on 3 June.
"It could have been worse, he's feeling better, the damage is not that bad," Arteta said when previewing the first leg of Arsenal's Europa League quarter-final tie against Slavia Prague.
"We have to go day by day to see how Kieran is feeling. Depending where we are and how he's feeling, we'll make this decision.
"It will depend on how things evolve. He is keen to play for us again this season and obviously his country in the Euros."
Tierney's injury is particularly problematic as Arsenal do not have another out and out left-back in their squad, with Sead Kolasinac having left for Schalke in January and Arteta unable to find a suitable replacement.
"We're going to have to make some adjustments. His qualities are unique. We don't have anyone to replace these qualities," Arteta added.
"We're going to have to find different ways to fulfil that gap and use all the things that can be as effective as well.
"We could not do anything in that window. We let Kolasinac out on loan and it's a decision that we made."
Bukayo Saka could be one solution to fill the void a little deeper down the left, with Arteta saying: "That's the reality: you have to change something, find a different way of attacking, find a different structure.
"It has to be stable and done by players who are doing that. We knew that position was a weakness around the squad. We have to find different solutions."
Saka himself has been contending with a hamstring issue but he, Emile Smith Rowe (thigh/quad) and Granit Xhaka (illness) are all in training in the hope of featuring, while Martin Odegaard will be assessed having struggled with an ankle problem since returning from international duty with Norway. David Luiz is still recovering from a knee injury.
The match against Slavia is particularly important for Arsenal, who put in an abject showing against Liverpool and know their most realistic chance of returning to European football next term is winning the Europa League having slipped to 10th in the Premier League.
Arteta feels the "big punch" inflicted by Liverpool may ultimately be a good thing for his side.
"I cannot criticise [the] hard work or ethic of my players, it's not about they don't want to run or try. There are areas where we are inconsistent, my demands are to be hitting those goals, achieving our best level every day, whether it's training or game day," he said.
"I didn't see anything different, that's why it was a big shock what happened against Liverpool, it's part of the journey, sometimes a big punch or slap in the face is good to return to reality.
"This competition is all or nothing because when you play a tie you are in or out. We know how important the game is for us."