The Spanish midfielder moved to Anfield earlier this month in a £25m transfer from Bayern Munich and made his debut in the 2-0 win over Chelsea.
He was omitted from the squad which beat Arsenal 3-1 on Monday night, and the club have confirmed it was due to contracting the coronavirus.
In a statement the club said: "Liverpool midfielder Thiago Alcantara has tested positive for COVID-19 and is currently self-isolating according to the necessary guidelines.
“The 29-year-old, who missed Monday’s match against Arsenal, has exhibited minor symptoms of the virus but is in good health and is getting better. 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 club doctor Jim Moxon said: “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 is expected to return to action following the upcoming international break.