Having already made his senior debut for Argentina, the 21-year-old has been linked with Manchester United over the past 12 months.
However, a switch to a smaller club now appears more likely, with West Bromwich Albion said to be one of the favourites for the attacker.
According to Football Insider, the Baggies may face competition from Leeds during the next window.
As a result of Gaich's £12m release clause, that battle is likely to be dependent on both clubs earning promotion to the Premier League.
Inter Milan, Atalanta BC and Genk are also said to be keen on the prospect, who has netted seven goals in 27 games for San Lorenzo.