Callaghan, 32, is Ms Thomas' former partner and the father of her five-year-old son. He appeared in court earlier this week and was remanded in custody.
The Herald on Sunday reported that Tanith Butler, who is eight months' pregnant with Callaghan's child, has been interviewed by detectives. She has had to vacate her house while it is searched by police, the paper reported.
Ms Thomas' son, Jack, lived with Ms Butler and Callaghan after his mother went missing, but has been cared for by other family members since his father's arrest on Tuesday.
On a Facebook page dedicated to finding her, Ms Thomas' mother, Teresa Scott, said Jack would be going to live with her in South Africa, where Ms Thomas was originally from.
Yesterday, police were at an inner city construction site in Victoria Street where searchers were lowered down a deep hole strengthened by a steel sleeve.
Items were also removed in bags as police excavated deeper into the hole.
Callaghan had worked at the Manson TCLM site.
It was understood a boat belonging to a friend of Callaghan had been tested and police confirmed numerous properties around Auckland had been searched.
Ms Thomas was reported missing on July 13 and her car was found in Hamilton. It had blood inside which police said suggested she had been murdered.
Police refused to say what they believed happened to Carmen other than she had been murdered.
They confirmed they were acting on information they had received, but did not know where her body was.
