Jose Mourinho sent one of his scout to Goodison Park to watch the pair take on Everton. Southampton lost 1-0 but it did little to dampen Chelsea's interest in two of their most prominent stars.
The Blues are currently compiling a shortlist of potential targets with the aim of strengthening their options ahead of what is likely to be a Premier League title defence. Mourinho is happy with his squad, but knows that failure to add extra competition to a squad can lead to complacency.
Making up the numbers
Both Schneiderlin and Clyne would likely arrive at Stamford Bridge as squad options but would certainly increase the overall quality of the squad. Clyne would compete with Branislav Ivanovic while Schneiderlin would have Cesc Fabregas ahead of him.
Clyne is about to enter the final year of his contract and unless Ronald Koeman can convince him to extend, will probably be pressured by Southampton officials to cash in. Manchester United are also interested in the England international and would be able to offer Clyne a more prominent role in the squad.
Louis van Gaal is looking for a quality right-back to replace Rafael, who looks set to leave Old Trafford in the summer.
Schneiderlin has indicated that he would want to leave St Mary's for a Champions League club but Southampton are under less pressure to sell the Frenchman. Southampton managed to keep him out of Tottenham's clutches last summer and only an offer in the region of £27m will convince the Saints to sell.
The same report suggests that Schneiderlin would represent better value for money than Paul Pogba, who is valued at £75m by his club Juventus. A large number of elite clubs will be chasing the youngster this summer and the price could stretch beyond the current world record.
It is likely that Chelsea will only have Arsenal competing for Schneiderlin's signature. Both will likely be able to offer him Champions League football, but it will be down to the role on offer.
Arsene Wenger has Francis Coquelin performing well in midfield and the likes of Jack Wilshere Aaron Ramsey and Mikel Arteta means playing time will be at a premium. However, it may prove an easier task to break into the Gunners squad than Chelsea's.
Are these the right targets for Chelsea or should they be chasing world-class players like Pogba? Give us your opinion in the comment box below!