The England international has not featured in either of Liverpool’s last two games after suffering the problem this month.
Sturridge, who missed a large portion of last season due to injury and underwent hip surgery over the summer, could yet be included in the squad to face the Saints at Anfield on Sunday.
However, Klopp insisted that the Reds would not rush the England international back to the team as they keep a close eye on his development.
Klopp told a news conference at Melwood on Friday morning: “I spoke a few minutes ago with [Daniel], it’s not much better at this moment.
“With Daniel’s injury we have to see day by day. If he can train tomorrow – if – completely normal, then he’s an option.
“Because the break is not so long now then not maybe as a starter but he’s an option [for the squad].
“Today we have another treatment, another scan, because there’s a little bit of fluid in the knee.
“We have to see but it’s day by day.”
Liverpool head into Sunday’s game looking to notch up their first win under Klopp, whose side were held to a 1-1 draw by 10-man Rubin Kazan in the Europa League on Thursday night.
The Reds drew 0-0 with Tottenham last weekend.
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