The two couples from Nakuru and Busia counties, who both claim they are the biological parents of a two-month old child, will wait longer for the DNA test results.
Police told a court on Monday that the results of the test ordered on August 13 and carried out on August 27 were not yet out.
Police took the baby from the couple in Busia on August 11 after the couple in Nakuru filed a complaint.
The child is currently in the custody of a children's home in Nakuru.
On Monday, Nakuru Principal Magistrate David Mararo said the case will be mentioned on September 30, when the results will be ready.
In their miscellaneous application in court, police sought to hold the woman from Busia, saying they were investigating her for child trafficking.
They requested permission to collect samples from both couples for the DNA test after the woman insisted that the child was hers.