Robertson was hurt during Sunday’s 1-1 friendly draw between the Reds and Athletic Bilbao at Anfield.
The Scotland international limped off after attempting to block a cross and was replaced at half-time by Owen Beck.
Klopp said further tests are required to determine the level of damage, casting doubt on Robertson’s availability for Liverpool’s opening game of the Premier League season away to Norwich City on August 14.
“We obviously don’t know in the moment how serious it is,” Klopp said.
“It was pretty painful and that’s why Andy went off. You probably saw it on the TV pictures better than we did, but he got the ball, the ball on the front of his foot, [and] twisted the ankle.
“The pain settled already, but we cannot say anything further without further assessment tomorrow [Monday]. He has to get a scan and then we will know more.”
Diogo Jota scored Liverpool’s opener on Sunday before Alex Berenguer levelled early in the second half in front of a crowd of close to 40,000.
There was a warm reception from the home fans for Virgil van Dijk, who played 72 minutes on home turf for the first time in over 10 months, having missed the majority of last season with a knee injury.
Liverpool face Osasuna on Monday in their final pre-season friendly before the 2021-22 season gets underway.
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