The France international only returned to training on Monday after Koeman labelled him as not being "phsically or mentally ready " to face Bayer Leverkusen in the Saints' final pre-season friendly.
Tottenham remain interested in signing the 24-year-old, and though Southampton claim they are not willing to sell another of their star players, Schneiderlin has made it clear that he would prefer a move away from the club.
But following a week of training, Koeman has revealed the midfielder could play a part at Anfield on Sunday.
"He [Schneiderlin] has had a very good week of training sessions," he told reporters.
"Last week he wasn't in shape to play. He's had a good week in training and that decision will be taken tomorrow [Saturday]."
If selected, Schneiderlin could line-up alongisde Shane Long after the Republic of Ireland striker completed his move to the club from Hull City on Thursday.
The south coast outfit paid out £12 million for the former West Brom forward, and Koeman revealed that a decision on Long's inclusion will be taken after training on Saturday.
"We are very happy that we signed Long because from the beginning of pre-season he was a target on our list to bring in.
"We lost some good players but brought in new ones. Long is an important player, we need that type of striker because we don't have that type in our squad.
"Tomorrow we have our last session and we'll decide on that [Long's selection] tomorrow."