The 24-year-old Armstrong netted 28 in 40 games for the Championship side last term.
Speaking at a press conference, Mowbray said: "Everything's agreed, I'm not sure there's not much to prevent that deal going through.
"Adam's fit, he'll be doing his medical. Things moved in different directions after the last few days, ultimately as I said after the game the other day it's only right that the club has valuation and it needed to be met.
"Southampton are the team that ultimately came up with the deal that was right for this football club. I wish Adam well, he's been in my office this morning.
"As he went round the whole club, the kitmen the cleaners, the kitchen staff, everybody, and said his goodbyes and thank you's, they all wished him well, he's been brilliant for this football club."
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