Arsenal have been confident of landing Schneiderlin in the summer and are sure to make a bid for the Frenchman at the end of the season.
But having watched the 25-year-old in action at Goodison Park, Chelsea are weighing up whether to make a rival offer and leave Schneiderlin with a difficult decision to make.
Ronald Koeman can expect bids for many of his best players this summer
Schneiderlin had an excellent game in Chelsea’s 1-1 draw with Southampton at Stamford Bridge in March.
Having rejected a £10million bid for Schneiderlin from Tottenham Hotspur last summer, Southampton are aware that they are unlikely to hang on to him this time around with Champions League qualification now looking unlikely.
Southampton are likely to demand around £25m for Schneiderlin, who last month reiterated his desire to play in the Champions League by saying: “In August, I wanted to take a move but the club had lost too many players to agree to sell me.
“Since then I have had discussions with the leaders and I want to play at the highest level: it is important at 25 years old to finally know the Champions League.
“Given that, late in the season, Southampton may qualify for the Champions League and I can stay, who knows. I have no desire to be in front of my TV on Tuesday and Wednesday.”
Other than looking at his midfield, Mourinho also wants to sign at least one new defender with Real Madrid’s Raphael Varane at the top of his list.
Chelsea are also interested in England right-back Clyne, which would allow Branislav Ivanovic to move inside and play more as a central defender.
Clyne’s current Southampton contract only has just over a year to run, which has sparked interest from United, Liverpool and now Chelsea.
Mourinho has made it clear that he is looking for a few key additions this summer, rather than any major overhaul. One big player who could leave is Oscar, who, together with Ramires, is attracting interest from Juventus.
“Everything starts with the squad we have and the players we want to keep,” said Mourinho. “And in that regard there is not a big space for new faces. The new faces will be a couple and will be a consequence of what the market can dictate.
“The team is good, the players are good, basically good. I try to be loyal to players. They deserve that loyalty for what they're giving. Even if direct competitors come in with huge amounts to buy top players, if my squad gives everything and gives us the Premier League title, loyalty will be stronger than a desire to buy new players. I would think the squad would be very, very similar to this one.”