The Daily Express claims Saints boss Mauricio Pellegrino, although it's more likely to be other figures at the club, has been monitoring the situation all summer as the midfielder now has less than a year left on his current deal.
Talks to tie Armstrong down to a new long-term deal began in early June but the 25-year-old has yet to reach an agreement with the Premiership champions over an extension.
Celtic boss Brendan Rodgers is desperate to keep the Scotland star at the club and remains hopeful negotiations can reach a positive outcome.
Speaking in July, Rodgers insisted Armstrong - previously linked to Brighton and West Brom - wants to stay at Parkhead.
He said: "We hope he signs a new one.
"I speak regularly with the boy and he wants to be here."
Reports on Friday claim the Hoops could be tempted to cash in on the ex-Dundee United star later this month instead of losing him for nothing next summer.