The 18-year-old forward is the said to be the next big thing to come out of the Saints’ famous academy, and caught the eye with his first Premier League goal in the weekend win over Norwich.
Gallagher has been slowly integrated into the Saints’ first team this season, with Mauricio Pochettino keen to keep him under the radar.
But it seems Chelsea scouts have already targeted the 6ft 4in striker, and are looking to bring him to Stamford Bridge at the end of the season.
Southampton have produced the likes of Theo Walcott, Alex Oxlade-Chamberlain, Adam Lallana, Gareth Bale and Luke Shaw in recent years, and the Blues reckon Gallagher could be the next big thing.
Chelsea are reportedly planning a bid of £5million, after Jose Mourinho was given the green light to try and launch a move this summer.
Southampton will look to resist any offer from the Blues for the England Under-19 striker – who is also eligible for Scotland – but Chelsea remain confident they will get their man.