Gallagher is considered one of Southampton's brightest prospects and Derby and Wolves are both said to be hoping the prospect of first-team football could persuade the Premier League club to let him go.
The report claims the Championship duo both want to sign Gallagher on loan, as they feel Southampton may be persuaded to allow the 19-year-old to leave on a temporary basis.
Gallagher has still not made an appearance under Ronald Koeman and there is a feeling that he may need to leave St. Mary's to get regular competitive football.
Southampton are set to invest heavily in their squad this summer and that could block Gallagher's pathway into the starting line-up for the time being.
The Saints still believe Gallagher has a future with them, hence Derby and Wolves are only trying for a loan deal, but a temporary spell could be good for his development.
Derby and Wolves are two of the most attacking teams outside the top-flight and Gallagher should thrive in a team which creates so many chances if he is to move.