The 23-year-old is about to enter the final year of his current contract with the Toffees and with speculation rife over his future, manager Ronald Koeman has given him a one-week ultimatum to commit his future or be sold on this summer.
The England international proved to be the difference last night with a 25-yard strike in the 1-0 win over Watford and Schneiderlin, who moved to Goodison in January, was full of praise for his "quality" teammate.
"We know the contract is between him and the club but what we can say is he is still working hard, has a good attitude and is very happy in the dressing room," he told Sky Sports News.
"We will see what is going to happen. We can't say too much on whether he will sign the contract or not - we want him to sign but it is his decision.
"Hopefully he will make a good one by signing the contract."
London rivals Arsenal and Tottenham Hotspur are among the clubs said to be chasing Barkley this summer, with reports suggesting that he could attract bids as high as £40m.