Dalglish was admitted to hospital on Wednesday for treatment of an infection which required intravenous antibiotics.
The 69-year-old former Celtic and Scotland footballer was routinely tested for coronavirus after being admitted.
"Unexpectedly, the test result was positive but he remains asymptomatic," the Dalglish family said.
Dalglish won the Scottish league title four times before moving to Liverpool in 1977. At Liverpool, his honours included eight league Championships as a player and manager and three European Cups.