According to Gavin McAffery of The Independent, while the Scotland skipper is not part of the squad for their opening UEFA European Championship 2020 qualifying match on Thursday, the left-back may be available for his country's match with San Marino on Sunday.
Celtic starlet Kieran Tierney is said to be set to replace Robertson on the left of the Scotland defence.
David Lynch of the Evening Standard provided more details on the nature of the surgery Robertson will have:
The Scot started for Liverpool in the Premier League against Fulham on Sunday, when a late goal from James Milner was enough to give the Reds a vital 2-1 win.
The victory saw Liverpool move back to the top of the table, two points clear of Manchester City, who have a game in hand.
After the match, Robertson shared the following post on social media, having given his shirt away to a travelling supporter:
"It was massive for us," he told the club's official website after the win. "It would have been a bit gutting had we not got the three points, if I'm honest. The first half, we dominated them and even in the second half we were very much in control up until the unexpected mistake."
Robertson has been a key part of Jurgen Klopp's side and has established a reputation as one of the best left-backs in the division.
The manner in which he raids forward gives Liverpool a constant outlet on the left flank. Robertson links expertly with Sadio Mane ahead of him and provides dependable deliveries into dangerous areas.
Robertson not being involved in Scotland's upcoming clash will be a major. Aside from Kazakhstan and San Marino, Scotland will face Belgium, Russia and Cyprus in their qualifying group.
Liverpool are next in action on March 31, when they face Tottenham Hotspur in a crucial contest at Anfield.