The Bayern Munich attacker is the highest-capped player in the squad, moving clear of Jurgen Klinsmann as the sixth-most capped player in Germany's history with his 109th appearance last time out against Liechtenstein, scoring twice in the 9-0 win.
Muller has worn the armband for Germany on a handful of occasions previously, but this would be the first time under new boss Flick, who believes the 32-year-old's presence on and off the field make him fully deserving of the honour.
"Thomas isn't someone to push himself into the spotlight; he prefers to support the team and fire the lads up," Flick said ahead of the game against Armenia. "He is also incredibly important off the pitch because he pushes the group again and again.
"He's a really important player for us, and he'll captain the side against Armenia. I'm happy to have him as a player because he's very, very valuable."
Regular captain Manuel Neuer will be rested and therefore has not travelled with the squad, missing out alongside Marco Reus, injured pair Leon Goretzka and Julian Draxler and the suspended Antonio Rudiger.
As a result, Muller is given the chance to captain the side, while Marc-Andre ter Stegen has the opportunity to feature between the sticks as Germany looks to end its FIFA World Cup qualifying campaign strongly, having won eight of its nine games so far to sit nine points clear at the top of Group J.
"Marc-Andre ter Stegen and Kai Havertz will both start the match," Flick continued. "We have already shown in the last few games what quality and mentality the team has.
"Armenia are very strong on the counter-attack, so we'll need to be very alert in defence. Our aim is to end this year with a win, of course."
Defender Christian Gunter is also relishing the chance to stake his claim for a spot in Flick's final FIFA World Cup squad next year.
"For me, it's about putting in a good performance for the team," Gunter said. "I'm part of a real competition for places and I want to give Hansi a tough decision when he comes to name his national team squad."