A La Liga statement read: "Among La Liga Santander and La Liga SmartBank (Segunda) clubs, five positive cases were detected in players, all of them asymptomatic and in the final phase of the disease".
In and official statement, La Liga said that they would not be revealing the identities of the players but Remiro announced the news in a post on Twitter.
La Liga's training plan includes players initially carrying out work individually before then expanding into small groups and finally taking part in larger team sessions.
La Liga logo is seen on the ball of the Spanish league football match between Club Atletico de Madrid and Levante UD at the Wanda Metropolitano stadium in Madrid on 15 April 2018.
Tebas noted the "very low" number of positive cases and said it will be virtually impossible for contagion to happen during games because everyone involved will have been tested ahead of time.
"LaLiga will continue to apply the Return to Training protocol approved by the CSD and Spanish Ministry of Health, in order to guarantee as much as possible the health and safety of all players, coaching staff and club employees". When they return two negative results, 72 hours apart, they will be able to join their teammates in training.
Tebas said that La Liga and their broadcast partners are testing various options to compensate for the lack of atmosphere, including interactive elements for viewers at home.
La Liga President Javier Tebas earlier said that he would like the season to resume from June 12.
The league reported on Sunday that five players across the two divisions had tested positive for the virus and had been ordered to stay isolated at home.
It added that all the players who have been found to be coronavirus positive are symptomatic.
La Liga will not confirm the names of any players who test positive, citing data protection laws.
At Madrid's Valdebebas training ground on Monday, players arrived in two shifts.
Barcelona was among the first teams to resume individual training on Friday.
Players will stick to the same individual physical training routine they have been following until now, in line with the club's instructions.