The first suggestions that the two clubs were close to finalising the transfer of the defensive midfielder emerged last month in the Spanish language version of OneFootball and from Sebastián Srur of Argentine outlet Radio Continental.
That was followed by reports that Sosa's agent would fly to England to conclude negotiations.
Sosa is currently on duty with Argentina at the U20 World Cup in Poland and he has admitted that he while flattered to be the subject of interest from Everton, his focus was on the tournament.
It was the Liverpool Echo's contention that while the Blues were keeping tabs on Sosa, they had no imminent plans to sign him and would likely revisit their interest in January but Diario Olé are saying that the transfer is very close to being concluded — Es Tiempo de River say it is done — but would be dependent on the player being granted rights to play in the Premier League.
Given that he has yet to be capped by Argentina's senior side, Sosa would have to qualify under the FA Exceptions guidelines.
His father, Juan, had this to say when he spoke with Tablón Radial where he appears to suggest that Santiago expects to receive an EU passport within the next few months:
"Santiago has been at River since he was 10 years old. We have mixed feelings; on the one hand we do not want him to leave and on the other it is a great [opportunity] for him.
"I know Gallardo talked to him, but he did not tell me what. The idea is that once you finish your participation in the World Cup, come to Buenos Aires and then travel to Liverpool to sign your contract. I will accompany him.
"Santi is missing a work visa. He is a young prospect. The community passport will be obtained by the end of the year.
"He has already spoken with the sports director and [coach] Silva knows him and wants him".