Two new cases of coronavirus, at two separate clubs, have been found after the latest round of Premier League tests.
Tests took place on Tuesday, Thursday and Friday of last week, with 996 players and club staff tested overall.
Players or club staff who have tested positive will now self-isolate for a period of seven days.
The number of tests available to each club was increased from 40 to 50 for this round of testing.
On 19 May the Premier League announced that in the first round, when 748 players and club staff were tested, there were six positive results, from three clubs.
Squads started non-contact training from Tuesday for the first time since the Premier League was suspended on 13 March because of the coronavirus pandemic, with 92 fixtures remaining.
Speaking on Friday, Premier League chief executive Richard Masters said the league was “as confident as we can be” about restarting in June.
Asked what date the Premier League was targeting for a return, Masters said it must be “flexible” and could learn from the resumption of Germany’s Bundesliga.
On Saturday the Spanish Prime Minister said La Liga can resume behind closed doors from 8 June.