Everton are keen on the Scottish international midfielder and could yet respond to any bids submitted for the 25-year-old.
Armstrong has just a year remaining on his contract at Celtic, and although Brendan Rodgers wants to keep the midfielder, there is a fear he could depart this summer.
Celtic have been unable to agree new terms with Armstrong yet, with the former Dundee United man seemingly keen to assess his options.
Southampton are now considering making a £3 million bid for Armstrong, as they look to take the midfielder on the cheap.
The Saints feel that Armstrong could represent good value for money, but they are unlikely to get a clear run to sign the midfielder, with both Everton and West Brom waiting in the wings.
Armstrong has made six appearances for Celtic already this season, scoring twice and carrying over his strong form from last season.