Jefferson Lewis, 47, was recently released from prison and was staying at the Alice Springs home at the time of Sharon's disappearance late on Saturday night, police said.
"We do believe Sharon has been abducted," acting Commander Mark Grieve told reporters in Darwin on Monday.
Sharon was reported missing from her Old Timers Aboriginal town camp residence about 1.30am on Sunday after being put to bed.
The search for Sharon has included dozens of police officers assisted by a helicopter crew, horse, dog and drone units, Tactical Response Group members and emergency service volunteers.
Police said Mr Lewis had been staying at the home where Sharon lived and went missing at the same time.
"He is believed to have been in the area around that time and staying at the residence," Mr Grieve said.
"He currently remains one of the few people who were in Old Timers camp at the time and who has not made themselves known to police, nor have police been able to locate him."
The man was not in prison for child-related offending before being released.
However, Mr Lewis had a history of assault and domestic violence charges, Mr Grieve said.
He was not directly related to Sharon but there might be a peripheral link, the senior officer said.
"Considering himself and Sharon went missing at around about the same time, it certainly brings about those suspicious circumstances and we'd like to speak to him about that," Mr Grieve said.
"If he is involved, we certainly hope that no harm has come to Sharon."
Sharon's mother has been provided with support, he said.
He called for any information from the public that could help locate Sharon.
"Obviously, it's a terrible situation to have such a young child go missing," Mr Grieve said.
"We're over 24 hours now, so it'd certainly be my worst nightmare as a parent.
"We want to find Sharon healthy and we want to get her back to her family."
Major crime squad detectives have been sent from Darwin to Alice Springs.
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