Hadeer Jamil Alonazi, 26, from Rusholme, South Manchester was reported missing on July 2.
Police say Hadeer and her six-year-old daughter, originally from Kuwait and members of the Bidoon Arab minority, had travelled to the UK in November 2018 from Mitilini, on the Greek island of Lesbos and were welcomed as refugees.
Greater Manchester Police said they were treating Hadeer's disappearance as murder after locating her daughter, who was also reported missing, but later found safe and well in London, where her father lives.
GMP have now confirmed they have received a missing from home inquiry and can therefore not 'exclusively treat' the case as a 'murder inquiry'.
Detective Chief Inspector Liz Hopkinson, of GMP's Major Incident Team, said:
DCI Hopkinson added: "It is a sensitive inquiry. We have got a lot of inquiries going on, but we do need help. I would ask anyone who believes they may have seen something suspicious that may be linked to Hadeer's disappearance to come forward.
"Inquiries suggest that Hadeer was living at the property in Hall Road up until June 2019."
Two men, aged 40 and 43, have been arrested on suspicion of murdering Hadeer and since been released on bail.