The 28-year-old was offered a fresh four-year deal with the Saints that would have made him the best-paid player in the club's history, according to reports.
The England international, who has earned three caps for his country, has been in lengthy negotiations to extend his stay at St Mary's.
However, with the forward's current contract set to expire in 2022, Ings is believed to have rebuffed Southampton s latest offer.
The news has put a number of Premier League clubs on red alert - with Spurs (7/2), champions Manchester City (9/2), Chelsea (11/2) and Manchester United (8/1) heading the betting.
Ings made 99 league starts for the Clarets, scoring 38 goals, after being brought to the club by Eddie Howe from AFC Bournemouth a decade ago.
He netted 21 times in the Championship in 2013/14 as he helped fire Burnley back to the top flight.
Ings christened his first season in the Premier League with 11 goals before completing a move to Liverpool in the summer of 2015.
He would move to the South Coast, initially on loan, just three years later after a spate of injuries hindered his progress at Anfield.