The Premier League side are once again looking to replace some of their bigger names.
The Sun claims that Tello will cost around £10m while Recio will be available for around half that price, but with the trio of departures netting the club around £60m, the pair's fees are unlikely to be a stumbling block for the south coast side.
The Sun adds that despite interest from other Premier League clubs for the two players - with Swansea keeping tabs on Tello while Everton want Recio - the Saints are in the driving seat to land both men.
New Southampton boss Claude Puel
While Recio will help fill the hole that Wanyama has left behind, Tello is arguably versatile enough to take Pelle's place up front or Mane's place in a slightly wider position.
Though the 24-year-old has hardly been prolific in front of goal during his career, he has struggled to hold down a regular spot in the team at his previous clubs - which is understandable given the stature of those sides - so having the chance to cement a place in Puel's team may well allow the Spaniard to show exactly how much talent he possesses.