The 24-year-old sat out Blackburn’s season-opening 2-1 Championship win over Swansea on Saturday as his future continues to be thrashed out.
After a fine goalscoring return last season, the former England Under-21 international has reportedly attracted plenty of Premier League attention.
The PA news agency understands Southampton have now struck a deal with Rovers, with Armstrong set for his medical on the south coast on Tuesday.
Armstrong came through the ranks at Newcastle before leaving for Ewood Park on a permanent deal in 2018.
It is understood Newcastle are in line for a slice of the fee as part of his sale to Blackburn three years ago.
The move, reportedly worth up to £15million, comes after Saints sold Ings – who scored 12 Premier League goals last season – to Aston Villa last week.
Armstrong made 15 top-flight appearances for Newcastle earlier in his career but went on to spend time out on loan at Coventry, Barnsley and Bolton.
He was then on loan at Blackburn for the 2017/18 campaign, completing a permanent transfer the following summer.
Blackburn finished 15th last season but Armstrong ended the Championship campaign with 28 goals, only bettered by Brentford’s Ivan Toney.