The newspaper claims Saints are trying for Liverpool's Danny Ings - a move that, if successful, would open the door for Gallagher to leave on loan.
And while United have already been linked with a move for Gallagher this summer, reports of Blackburn's interest appear to be fresh.
The 22-year-old is no stranger to the Lancashire club, having impressed on loan there the season before last.
According to the Sun on Sunday, however, both Blackburn and Sheffield United could struggle to meet Gallagher's £24,000-a-week wages, meaning a temporary deal would have to be subsidised.
The 6ft 4in player, signed by Southampton from Plymouth Argyle as a youngster, has scored 17 goals in 89 games at Championship level.
Is the Southampton striker bound for Sheffield United, Blackburn or neither?
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