The 26-year-old was a key part of the Clarets squad which cantered to the Championship title last season, scoring 11 goals and providing three assists in 33 second-tier appearances.
However, the arrivals of Luca Koleosho and Wilson Odobert from Espanyol and Troyes respectively in the summer have led to Benson's game time being restricted so far in 2023-24.
The former Belgian youth international has featured in just 73 minutes of Premier League football this term, starting his only top-flight match in the defeat against Aston Villa on August 27.
According to Belgian journalist Sacha Tavolieri, Southampton are stepping up their attempts to secure the services of Benson as the January window enters its final fortnight.
The report claims that Russell Martin's side have submitted a second offer for the attacker as they look to lure the Clarets into a deal this month.
It is understood that Benson would move to St Mary's Stadium on loan for the remainder of the 2023-24 campaign, with the South Coast club possessing the option to make the switch permanent in the summer.
Struggling for game time at Turf Moor under Vincent Kompany, the winger is said to be keen on a move to the promotion-chasing Saints in the coming weeks.
Southampton sit just three points behind second-placed Ipswich Town in the Championship standings, currently enjoying a 19-game unbeaten streak in the second tier.
Martin's men are not the only side interested in Benson, though, with Championship rivals Hull City also believed to be aiming to secure the signing of the Belgian.
It was reported earlier this month that the Tigers had verbally
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