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The 24-year-old forward joined Union Saint-Gilloise last summer and has shone in Belgium, scoring 21 times in 33 matches.
He featured against Liverpool in the Europa League earlier in the campaign, scoring one and setting up another in his side's 2-1 victory back in December.
He has three years remaining on his current deal and it is claimed Union SG would be looking for $22m (£17m/€20m) to let him go this summer.
Liverpool is likely to face competition from the Premier League, if it does make a formal bid.
Amoura is far from the finished article but has shone at Union SG this season and caught the eye against Liverpool in the Europa League.
It remains to be seen if he is ready to play in the Premier League at this stage of his career.
Crysencio Summerville (Leeds)
Leeds winger Crysencio Summerville remains on Liverpool's radar, reports Give Me Sport
Dutch outlet 1908.nl reports that incoming boss Arne Slot is a big fan of his countryman but both Leeds and the player will wait until the end of the season before making a decision.
Liverpool.com says: Another winger to be linked with Liverpool, Summerville has shone in the Championship this term
He does have Premier League experience and will be eager to test himself again after Leeds' relegation last season