The 20-year-old has scored 33 times and registered nine assists in 70 appearances for his Uruguayan club in all competitions, including 23 goals and five assists in 40 outings during the current season.
Real Madrid were credited with an interest in the attacker earlier this year, with Los Blancos thought to be keen to sign him for their Castilla outfit with the view to progressing into the first team.
However, according to Mundo Deportivo, there is also genuine interest from Barcelona, who are looking for low-cost signings in 2022 due to their financial problems.
The report claims that an offer in the region of €12m (£10.2m) is likely to be accepted by Uruguayan champions Penarol.
Alvarez, who has scored once in four appearances for Uruguay, has a contract with his current club until June 2024. body check tags ::