The Argentine report claims that Liverpool have already had an $8 million [roughly £5.65m] approach rejected with the Buenos Aires-based giants instead looking for a bid in the region of £8.5 million.
Alario rose to prominence in Argentina following his move from Colon to River Plate last summer.
The 23-year-old striker made an immediate impact, helping the club win the Copa Libertadores by scoring the first of his side’s thee goals in a one-sided victory over Mexican side Tigres UANL.
Alario then went on to feature in the World Club Cup last December, scoring the only goal of the game as River defeated Sanfrecce Hiroshima 1-0 before going on to lose to Barcelona 3-0 in the final.
Despite not being capped by the Argentine national team, the 5ft 11inch striker has made plenty of noises over the past 12 months and Liverpool are among those reportedly keen on signing him.
The Reds have Daniel Sturridge, Divock Origi, Roberto Firmino and Christian Benteke as potential striker options yet with the latter out of favour, perhaps there is scope to look for a possible replacement.
Alario has ten goals in 19 league appearances for the South American champions, but it is his energy and feisty style of play which will have caught the attentions of European scouts.
Infobae goes on to say that Palermo, Inter Milan and Ajax are also monitoring his progress.