Southgate could have a selection headache for the final group match on Tuesday.
The midfield “experiment” of Trent Alexander-Arnold received criticism following the draw to Denmark.
BBC Sport currently understands that Gallagher is likely to come into the England starting XI in his place, ahead of other alternative midfield options such as Kobbie Mainoo or Adam Wharton.
Chelsea’s Gallagher has been a second-half substitute in both England’s games so far.
The debate rolls on around Phil Foden, so could Southgate opt for Anthony Gordon to increase the pace down the left-hand side, or even Cole Palmer?
Elsewhere, England left-back Luke Shaw returned to team training on Monday but is still not available for selection.
Kieran Trippier has been filling in for the injured Shaw, but should he choose to rest the natural right-back, BBC Sport understands that there is a strong possibility that Aston Villa defender Ezri Konsa could come into that position.
Slovenia’s star forward Benjamin Sesko is expected to start despite being withdrawn with a quad issue against Serbia on Thursday.