Robertson went off injured during Scotland's 2-0 European Championship qualifying defeat to Spain last Thursday.
The defender cradled his right arm in his shirt as he was guided off the pitch and he missed the 4-1 friendly loss to France this week.
Is Robertson fit to play for Liverpool this weekend?
Robertson will miss the game against Everton this weekend as he gets set for a lengthy spell on the sidelines.
Liverpool are awaiting scan results of his dislocated shoulder that will determine whether he will require surgery or not.
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According to the Mirror, Robertson could face up to 10 weeks on the sidelines if it's deemed he needs to go under the knife.
Even if the 29-year-old avoids surgery, he will still definitely miss the clash with Everton, with Scotland boss Steve Clarke expressing his doubt that his captain will be available for the Tartan Army's final two Euro 2024 qualifiers next month.
"As always with injured players, we send them back to their club; Andy's gone back to Liverpool," Clarke explained.
"Liverpool will deal with his injury. Obviously, it's a shoulder issue and Liverpool will deal with it. If he's fit for next month, it's great. If he's not fit for next month then hopefully he'll be fit as soon as possible for his club."
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The news will come as a huge blow to Liverpool, with Robertson playing every minute for the Reds in the Premier League this season.
The simplest change would be a like-for-like swap with his deputy Kostas Tsimikas and that may indeed prove to be what Klopp does.
But the Greek international has flattered to deceive this season in the Europa League and was directly at fault for Leicester's opener at Anfield in Liverpool's 3-1 Carabao Cup comeback win last month.
This may open the door for Joe Gomez to start at left-back instead, where he first broke into the Liverpool team under Brendan Rodgers.