Dalglish was admitted to hospital on Wednesday due to an infection which required intravenous antibiotics, but was also tested for Covid-19.
The former Scotland forward was told to remain in hospital for treatment despite being asymptomatic.
In a statement released by Liverpool FC, his family said: "In keeping with current procedures, he was subsequently tested for COVID-19 despite having previously displayed no symptoms of the illness. Unexpectedly, the test result was positive but he remains asymptomatic.
“Prior to his admission to hospital, Sir Kenny had chosen to voluntarily self-isolate for longer than the advised period together with his family.”
"He would urge everyone to follow the relevant government and expert guidance in the days and weeks ahead."
The statement added: "He would like to take this opportunity to thank the brilliant NHS staff, whose dedication, bravery and sacrifice should be the focus of the nation’s attention at this extraordinary time."
During his playing days in Liverpool, Dalglish won six Football League First Division titles, three European Cups, one FA Cup and four League cups, scoring 172 goals in 515 games for the Reds during in 13 years.
After his illustrious playing career, Dalglish coached Liverpool for two spells and won three league titles, two FA Cups and one League Cup.
