The £20m summer signing from Benfica suffered a hamstring strain while on the club's pre-season tour in USA, says the Daily Star.
Markovic, 20, will definitely miss tomorrow's friendly with Borussia Dortmund at Anfield.
And Kop boss Brendan Rodgers will be sweating on the attacking midfielder's fitness ahead of their Premier League opener against the Saints.
It is another blow for Rodgers who is already without £25m signing Adam Lallana for the first month of the campaign because of a knee problem.
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