The Spain international, who joined the club from Bayern Munich earlier this month in a deal worth £25million, is self-isolating at home after his diagnosis.
The 29-year-old has made just one appearance for the club so far when he came on as a substitute in Liverpool's 2-0 win against Chelsea.
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."
Liverpool's first-team doctor, Dr Jim Moxon, added: "It is the choice of each individual whether they reveal publicly the outcome of a Covid-19 test.
"We have followed all the protocols to the letter and Thiago is OK.
"He will be self-isolating now as per the guidelines and hopefully he'll be back with us soon."
Thiago had been in line to make his full debut against Arsenal in the Premier League on Monday night, a game they went on to win 3-1, but was a notable omission from the squad.
He will now also miss Thursday's Carabao Cup visit of the Gunners and Sunday's Premier League trip to Aston Villa.
His first start for the club is now likely to be in the Merseyside derby at Everton on October 17.
Manager Jurgen Klopp, speaking on Monday, said: "After the international break, he will be fine.
"For sure and that's the situation at the moment, he isn't fit enough to be available tonight and that's it."
Thiago Alcantara will miss Liverpool's next two games having tested positive for coronavirus.