Chelsea have added three players to Conte’s squad so far this summer, with Tiemoue Bakayoko joining Antonio Rudiger and Willy Caballero in signing for the Premier League champions.
But Conte is determined to push on in the transfer market after a quiet start to the summer and hopes to bolster his defensive options.
Brazilian Danilo has emerged as a top target and the Evening Standard claim Chelsea are close to agreeing a deal with Real Madrid to sign the full-back.
The 26-year-old can operate on either side of the defence and is expected to compete with Victor Moses for a place in the starting line-up.
But despite closing in on the transfer, Conte is considering making further defensive signings and is keeping tabs on Bertrand, according to the Standard.
After starting off in the Gillingham youth system, Bertrand signed for Chelsea in 2006 but only played 28 league games before being sold two years ago.
His future at Southampton is uncertain and Liverpool and Manchester City also expressed interest in a deal earlier in the year.
Conte is also desperate to sign a star striker and, having missed out on Belgium international Romelu Lukaku, is said to have turned to prolific Argentine Sergio Aguero.
However, reports say Manchester City have no interest in selling the 29-year-old this summer and will not entertain any approach from Chelsea.