As a result, Tierney will be forced out of Scotland's three upcoming matches, as well as Arsenal's Premier League encounter against Manchester City on October 17.
Tierney, though, has denied coming into contact with Armstrong and Arsenal have said they are seeking "further advice and clarification" into the situation.
The Scottish FA confirmed on Wednesday that Armstrong had returned a positive test after originally testing negative upon arrival on Monday.
The Southampton midfielder will self-isolate for 10 days while Tierney, Ryan Christie and two Scotland staff members will self-isolate for 14 days after being confirmed as close contacts with Armstrong.
In a statement released by Arsenal, Tierney expressed his frustration at the situation while claiming that he socially distanced from all his team-mates.
"I'm so disappointed and frustrated to be in this situation," Tierney said.
"I have adhered to all regulations and made sure I was socially distancing from my team-mates in the hotel. I've also tested negative. I know Arsenal and the SFA [Scottish Football Association] are now in discussions with the Scottish authorities to gain a further understanding."
Arsenal backed up their player while indicating they would look for a way to avoid being without their left-back when they visit the Etihad Stadium.
"Our medical team have confirmed that Kieran was socially distant at all times from the player who tested positive and has broken no rules regarding Covid-19 protocols. We are currently seeking further advice and clarification of the details," the Gunners added in their statement.
Scotland will be without Armstrong, Tierney and Christie for Thursday's Euro 2021 qualifier against Israel as well as UEFA Nations League matches against Slovakia on Sunday and the Czech Republic next Wednesday.
Tierney has started three of Arsenal's four Premier League matches in 2020-21 as the Gunners have started the season with nine points from those four games.