The government previously denied the patients are missing and tried to reassure people, saying they had been moved to another health facility.
The World Health Organization (WHO) has called for exit screenings on all travelers from affected countries.
It wants checks at airports, sea ports and major land crossings.
Several airlines have already stopped flying to Guinea, Liberia and Sierra Leone – the countries worst affected by the world’s most deadly outbreak of Ebola, which has no known cure.
The Ebola virus has killed 1,145 people this year, the WHO says.
Assistant Health Minister Tolbert Nyenswah said protesters in the West Point district attacked a quarantine centre on Saturday because they were unhappy that patients were being taken there from other parts of the capital, Monrovia.
Other reports suggested the protesters had believed Ebola was a hoax and wanted to force the centre to close.
Tolbert Nyenswah had said that all the suspected patients had been transferred to an Ebola treatment centre in the John F. Kennedy Memorial Medical Center in Monrovia.
However, on Monday, Liberia’s information minister said 17 of the 37 patients were unaccounted for.
He said the authorities were now trying to track them down but said he was confident they would return.
He said the attack on the quarantine centre was Liberia’s greatest setback since the Ebola outbreak began.
Blood-stained mattresses, bedding and medical equipment were taken from the centre.
The looting spree could spread the virus to the whole of the West Point area.
Lindis Hurum, from medical charity Medecins Sans Frontieres (MSF), says there is an urgent need for a massive public awareness campaign in Liberia.
MSF says the Ebola outbreak has had a terrible impact on Liberia’s entire healthcare system, which it says is more or less falling apart.
Many health facilities have closed, with patients as well as medical staff, too scared to turn up for fear of catching the disease.
The Ebola epidemic began in Guinea in February and has since spread to Liberia, Sierra Leone and Nigeria.
One Nigerian doctor has survived the disease and was sent home on Saturday night, said Health Minister Onyebuchi Chukwu in a statement.
Onyebuchi Chukwu said five other people infected with Ebola had almost fully recovered.
The death toll of 1,145 was announced on Friday after the WHO said 76 new deaths had been reported in the two days to August 13. There have been 2,127 cases reported in total.
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