They had claimed earlier in the week that Wigan were interested but now believe Blackburn will submit an official bid for the 23-year-old.
Gallagher has spent time away from St Mary’s over the past few seasons with loan spells at Birmingham, Blackburn, and MK Dons. He did remain with the club last season, however, he failed to make an impression in the first-team and his four appearances all came from the bench.
The arrival of Che Adams this week means there is a surplus of forwards at the club and Gallagher would most likely find himself even further down the pecking order.
He joined the club as a trainee but, of his 22 Premier League appearances, 19 of them have been as a sub. He is valued by Transfermarkt at £2.7million and has two years remaining on his contract, but it would appear his future lies elsewhere.
Sky Sports have stated Ralph Hasenhuttl wants to trim his squad and move on players to give him greater flexibility in the wage bill to bring new players in.