The club's penultimate preseason match was dampened when Robertson, 27, appeared to sustain a serious ankle injury before half-time which saw him replaced.
"We obviously don't know in the moment how serious it is," Liverpool boss Jurgen Klopp told a postmatch news conference. "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. He has to get a scan and then we will know more."
On Monday, Robertson tweeted an update saying: "Thanks to everyone for the kind messages and support. Scan suggests nothing too major but there's some ligament damage which will need to mend. I will be grafting every day so I can help the team again sooner rather than later. Good luck to the boys playing tonight."
The match was played in front of Anfield at 75% capacity -- the biggest crowd at the ground since the start of the coronavirus pandemic.
Kickoff was delayed by 30 minutes after thousands of supporters queued outside to enter the stadium.
There was good news for Liverpool as Virgil van Dijk started the game and received a standing ovation when he was substituted after 71 minutes.
Liverpool take on Osasuna in their final preseason match before they begin their Premier League campaign away at Norwich City on Aug. 14. Athletic face a trip to Elche on Aug. 16 for their first league match.