Brazilian striker Leo Bonatini arrived on a season-long loan from Al Hilal yesterday as Wolves landed their first new striker of the summer.
But with Nouha Dicko – currently out injured – their only other senior out-and-out striker, the club hope to bring in another before the August 31 deadline.
Gallagher is a preferred option although Wolves face tough competition from other Championship clubs – and sources suggest any move may depend on the Saints bringing in another striker themselves, so any prospective deal isn't close.
Charlie Austin, Manolo Gabbiadin, Shane Long and, if utilised up front, Nathan Redmond are ahead of Gallagher in the pecking order for Mauricio Pellegrino's team.
However Gallagher has travelled to Southampton's pre-season tour in France. He came off the bench to play six minutes in their 3-0 win over Saint-Etienne on Saturday.
The Express & Star understands that a loan move is preferable for whoever Wolves bring in due to budget constraints and concerns over breaching Financial Fair Play regulations.
They have spent £19m this summer on Ruben Neves, Roderick Miranda and Barry Douglas, in a total of £47m on transfer fees alone over the past 12 months, although that doesn't include loan fees.
Wolves already have four loanees – Willy Boly, Diogo Jota, Ruben Vinagre and Bonatini – on their books this season. EFL rules allow an unlimited number of loan players to be signed, but only five can be named in a matchday squad.
Gallagher, 6ft 4in, spent last season on loan at Blackburn Rovers and scored 12 times in 46 appearances – including one against Wolves – in a team that was relegated to League One.
The England Under-20 international came through the youth ranks at Plymouth Argyle before making the switch to Southampton aged 16.
He signed a new four-year contract with the Saints earlier this month.