Tierney was forced to limp off during the 3-0 defeat at Emirates Stadium and Arsenal’s initial fears were that he’d be out for a few weeks.
The left-back has undergone further assessment and he will be out of action for up to six weeks, but he does not require surgery.
When questioned on the injury on Wednesday, Arsenal boss Mikel Arteta has refused to say whether Tierney will play again this season
"We have to make sure first that he doesn't need any surgery, which it doesn't look like," he said.
"The timeframe will be that. It could have been worse because the action looks quite scary, but he looks a bit better and the damage was not that bad, so at the end of the day it is not the bad news we could have been waiting for.
"We are going to have to go day by day and see how Kieran is feeling, then when he gets close to playing time and how he is feeling we will make that decision."
Asked if he thinks Tierney will be fit for Scotland at the rearranged European Championship finals, Arteta added: "It will depend how things evolve again in the next few weeks."
Tierney’s injury is a huge blow to Arsenal and Arteta, with the 23-year-old the only out-and-out left-back in the Gunners’ squad.
In his absence, Cedric Soares or Bukayo Saka are the most likely candidates to fill in there.
Arsenal will be without Tierney and David Luiz for Thursday’s Europa League quarter-final with Slavia Prague - and Martin Odegaard could miss out too.
The attacking midfielder has been experiencing discomfort to his right ankle since being on international duty and is being assessed.
In better news, Emile Smith Rowe (hip), Granit Xhaka (ill) and Saka are all fit to return for Thursday’s first leg at Emirates Stadium.