The 29-year-old, who joined from Bayern Munich this month, will now self-isolate.
A Liverpool statement read: “Thiago Alcantara has tested positive for Covid-19 and is currently self-isolating according to the necessary guidelines.
“The club has, and will continue to follow, all protocols relating to Covid-19 and Thiago will remain in self-isolation for the required period of time.”
“After the international break, he will be fine,” the German told Sky Sports.
“For sure and that’s the situation at the moment, he isn’t fit enough to be available tonight and that’s it.”
Thiago signed for a reported £20million, rising to £25million, after seven years at Bayern, where he won seven league titles.
His final game for the Germans was their 1-0 Champions League victory over Paris St Germain in August.
He will miss Thursday’s Carabao Cup visit of Arsenal and Sunday’s Premier League trip to Aston Villa but could be available for the Merseyside derby at Everton on October 17.