Spurs have agreed a £15million deal for the Danish midfielder.
Right-back Kyle Walker-Peters is set to move the other way in a separate deal believed to be worth about £12m, so Tottenham are effectively paying £3m plus Walker-Peters for Hojbjerg.
Everton also had interest in Hojbjerg, but the 25-year-old, who joined Southampton from Bayern Munich four years ago, made it clear he wanted to move to north London.
Jose Mourinho has identified Hojbjerg as a key transfer target and the deal could be completed this week.
After securing a sixth-placed finish and a Europa League spot for next season, Mourinho is looking to strengthen further this summer with defensive reinforcements.
Walker-Peters was on loan at Southampton this season and will make the move to St Mary’s permanent.
The 23-year-old has made 24 appearances for Spurs since making his debut under Mauricio Pochettino in August 2017.