The Sun says Alario, 23, is keen to play in the Premier League, with Jurgen Klopp's Liverpool and Mauricio Pochettino's Spurs his most likely destinations.
The forward has a £15.2m ($20.1m) release clause in his contract with River Plate, which runs until 2020.
He made his international debut for Argentina in the World Cup qualifier win over Uruguay on 1 September and also featured in the 2-2 draw against Venezuela on 6 September.
Serie A clubs Inter Milan and Torino are also interested in Alario, according to the Sun.
A Premier League move for the striker is bound to be complicated as third-party ownership issues will have to be dealt with before he can move to England.
Premier League interest
Liverpool have been monitoring Alario for some time and are reported to have had a previous bid of £8.5m rejected.
The player's agent revealed in July that Spurs were interested in signing the 23-year-old.
"Today Alario is at River. There was only one proposal, from Genoa, and interest of Tottenham. If Tottenham pays the clause, Alario is going," Pedro Aldave was quoted as saying by ESPN.
Alario said earlier in the summer that it was his "dream" to play in Europe.
"My representative did not say anything concrete on the subject and now I think of River, to get back on the pitch and give back everything this club gave me," he told Argentine newspaper Ole.
"Like every player, I dream to play in Europe at some point in my career. For that first I have to make things right at River, to keep growing and learning."
Alario made 26 appearances for River Plate last season, scoring 11 goals. He played over 50 league matches for Colon between 2011 and 2015 before joining the Buenos Aires giants.