One of the mothers, Rashmita Mallick, had claimed that she had given birth to a boy on March 30 at the hospital and had refused to accept the girl she was handed over.
Another mother, Nirupama Mallick of Jajpur district, delivered a baby around the same time and had left for home with a boy.
Seeking a DNA test, Rashmita`s husband Sushanta Mallick had filed a writ petition in the High Court claiming that his wife had delivered a son and damages for trauma suffered because of the alleged baby swap.
The high court on Tuesday had ordered a DNA test of the babies in the presence of a designated court officer to determine their parentage. Though Rashmita was present in the hospital with her baby on Wednesday for the test, Nirupama was not.
Hospital authorities said another date would be set for the DNA test.