The Sunday Post claim that Armstrong was heavily linked with a move to the Saints during the summer transfer window and they’re now ready to rekindle their interest.
The Saints enquired about the player’s availability in the summer, but were given a firm no by the Scottish Champions, who were focusing on their Champions League qualifiers at the time.
The report suggests that Celtic could consider selling the player in January, despite still being involved in the Europa League, with a tie against Zenit St Petersburg to come.
The 25-year-old has been in and out of the Celtic side this season, making just 12 starts, scoring twice in the league.
Armstrong joined Celtic from Dundee in 2015 as part of a £2 million deal that saw him and Gary Mackay-Stevens join the club.