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The 23-year-old full-back limped off before half-time in a 3-0 Premier League loss at home to Jurgen Klopp's men on Saturday.
Arsenal confirmed on Wednesday that Tierney suffered knee ligament damage during the game and could be sidelined for the next six weeks.
If that is the case, he could return for the final league game of the season at home to Brighton on May 23 -- with the Europa League final taking place three days later.
But Arteta, speaking ahead of Thursday's Europa League quarter-final first-leg tie against Slavia Prague at the Emirates, was keen not to make promises over Tierney's immediate future.
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"We have to make sure first that he doesn't need any surgery, which it doesn't look like," he said.
"If that's the case, the timeframe will be around that (six weeks). It could have been worse because the action looks quite scary but he's feeling better and the damage is not that bad.
"At the end of the day, it's not the bad news that maybe we were expecting after the game."
Arteta also stopped short of confirming whether he felt Tierney would be fit for Scotland's Euro 2020 campaign, adding: "Again, it will depend on how things evolve in the next few weeks.
"He's so keen to play for us before the end of the season and he's extremely keen to represent his country in the Euros."
The Euro 2020 tournament, delayed by a year because of the coronavirus pandemic, starts on June 11.
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