Photographs of the suspect, who some know by the nickname "Dole" began circulating on social media on Wednesday, before Joseph's body was recovered by fishermen in the Gulf of Paria.
The man's home was searched by officers on Tuesday.
Police said he was a close friend of the 39-year-old man who was detained on Friday.
The decomposing body of the mother of one became tangled in a fishing net at the mouth of the Caroni River on Wednesday.
Reports say The 24-year-old woman, who joined the TT Police Service last November, had her hands tied behind her back and a stone secured in the knot; had two large stones and a steel anchor secured around her neck via a length of rope and had the right side of her face bashed in.
The T&T Coast Guard took the body to the Tobago Ferry Terminal and then removed to the Forensic Science Centre in St James.
An autopsy would be performed this morning to determine how she died. Joseph, 22, did not return to her Morvant home after leaving for work last week Thursday. She joined the police service last November.
A missing person report was filed on Friday. A candlelight vigil was held in Joseph's memory at the Church of Epiphany in Morvant last night.