The report claims that both the Saints and the Seagulls are following the player closely ahead of the January window, with either side needing €25m (£22.4m) for Sociedad to part ways with the 25-year-old.
The Belgium international, formerly of Manchester United, can play on either flank and in an attacking midfield role.
However, according to the latest reports, the wide man could be in line for a possible return to England.
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David Moyes compared Januzaj to a young Wayne Rooney back in 2014 when in charge of United.
The Belgian’s possible St Mary’s arrival would hand Ralph Hasenhuttl another creative wide option.
His average league match rating has improved since leaving England, and now a full international, he appears to be approaching the prime of his career.
A move for Januzaj could turn out to be a shrewd one, however, the reported £22.4m price tag may put the Saints off.