Aarons only joined the Cherries last summer from Norwich City for a fee around £7m and made 23 appearances that have obviously impressed the Saints.
Rumours have sparked since Southampton were promoted to the Premier League and have since been looking to improve their squad.
Aarons has had limited game time at Bournemouth thanks to veteran defender Adam Smith taking his spot in the starting eleven.
Bournemouth rejected the £5million bid for Aarons so if the Saints are serious about the 24-year-old then they will have to make a much bigger bid which is higher than what the Cherries paid last season.
Saints' defender Kyle Walker-Peters has been heavily linked with an exit in recent weeks which explains Southampton’s desperation to sign fresh talent.
Aarons stated he was “looking forward to a really good pre-season now” but whether that is a pre season at the Cherries or the Saints remains to be seen.