Families have up to seven days to identify and collect the 541 bodies at the Kenyatta National Hospital mortuary, it said in a notice on its website.
"Failure to which the hospital will seek authority from the courts to dispose them," it said.
More than three quarters of the bodies are of children who died from the middle of last year.
While most of the names and ages of the deceased were published, the identities of more than a dozen adults was unknown.
Under Kenyan public health laws, no one is allowed to keep a body at a public mortuary for more than 10 days before burial.
A penalty is accrued for extra days beyond the allocated period and failure to comply with the rules can attract a prison sentence of up to six months.
Cases of unclaimed bodies are not uncommon in the East African nation where authorities say relatives are seldom aware of the deaths or they abandon their loved ones due to high hospital and mortuary fees.
Bodies that remain uncollected after the grace period are buried in mass graves.
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