Adams, 27, scored 16 Championship goals to help the Saints return to the Premier League.
His contract expires in June, and he is rumoured to join Wolves.
Southampton stated in a statement that discussion with Adams on an extension is ongoing.
Scotland has selected the £15m signing from Birmingham City in 2019 and Armstrong, who is leaving Southampton after six years, in their proposed Euro 2024 team.
Armstrong, 32, was a vital player in 2023-24, however he left as a free agent after missing the play-off final triumph against Leeds at Wembley on 26 May due to injury.
Armstrong and Adams are on Scotland’s Euro 2024 team.
“The club would like to thank Stuart for his commitment and effort during his time as a Saint, and we wish him the very best in whatever comes next,” said Southampton.
The club announced 34-year-old custodian Alex McCarthy is negotiating a new contract.
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