It is believed the Blues have already wrapped up deals for Andre Schurrle and Wallace, with the Saints youngster the next to arrive at Stamford Bridge.
Chelsea see Shaw as an excellent long-term replacement for Ashley Cole and they are eager to sign him this summer before his price increases any further.
The Blues could face stiff competition for the player’s signature from their Premier League rivals, but they do appear to be leading the chase at the moment.
Luke Shaw, 17, has only made his league debut this season, but he has quickly made the left-back position his own and went on to make 24 Premier League appearances.
The youngster has shown incredible maturity for his fragile age and it looks as though Southampton will not be able to keep hold of him in the summer transfer window.