Mitroglou scored three goals for Greece in their World Cup play-off win against Romania.
The Daily Mirror reports that Liverpool head of recruitment Dave Fallows and chief scout Barry Hunter were in Romania to watch the striker in action.
The pair also watched the first eg of the tie in Greece.
Liverpool scouts have reportedly watched the striker’s last six matches.
Liverpool boss Brendan Rodgers is keen to bring in cover for star duo Luis Suarez and Daniel Sturridge.
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