Hoops boss Brendan Rodgers confirmed at his pre-match media conference on Friday that the pair would not recover from injury in time to face Slovakia and Slovenia next month.
Brown limped off against Anderlecht on Wednesday with a tight hamstring while Armstrong sat out the Champions League game in Brussels.
Speaking a day ahead of Celtic’s Ladbrokes Premiership encounter with Hibernian, Rodgers said: “Scott has had a scan so him and Stuart Armstrong will not be available for a few weeks.
“It is unfortunate, just a slight strain of top of the hamstring and calf.
“It is unfortunate for both, unfortunate for us and obviously for Scotland.”
The news is a major blow to Scotland boss Gordon Strachan, whose decision to build his team around Rodgers’ successful side has helped rejuvenate the country’s hopes of reaching next summer’s finals in Russia.
Brown returned from a brief international retirement after missing the first three qualifiers while Strachan hailed Armstrong’s performance against Slovenia as the best Scotland debut he could remember, after the former Dundee United player set up Chris Martin’s late winner.
Scotland go into their final two games knowing that victory in each would secure second spot and a potential play-off place.
Strachan will now have to rejig his central midfield and could add reinforcements before his side meet up on Sunday ahead of Thursday’s Hampden clash against Slovakia.
Darren Fletcher is back in the squad after missing the victories over Lithuania and Malta through injury, while both James Morrison and James McArthur featured in those games.
Strachan could also utilise Barry Bannan and John McGinn in the middle.
Rodgers will now hope both players recover in time for Celtic’s third Champions League group game against Bayern Munich in Germany on October 18.
The Celtic manager said: “We don’t think it is obviously serious. There are various gradings on these types of injuries and it is something that is probably a few weeks.
“So it puts them out of the game tomorrow and the international games.”