Janet McQueen, 58, was last seen by a neighbour leaving her home in Langside Road in the Govanhill area of Glasgow at about 9pm on Tuesday 18 October.
Ms McQueen left her dog, phone and money in her flat, and there has been no activity on her bank account since her disappearance.
Police have carried out extensive investigations, including specialist search teams combing rear courtyards around her home and in nearby Queen’s Park, but they have found no trace of the missing woman.
Ms McQueen’s family are said to be “desperate” for her to come home and police urged anyone with information to contact them.
Inspector Alex Hutton, of Gorbals police station, said: “This is a horrible time for the family; they just don’t know where Janet is or what, if anything, has happened to her.
“The family kept in touch with each other and for Janet to be out of contact for so long is concerning and very, very distressing for them.
“Janet left her flat around 9pm on the Tuesday night and for some reason she left behind her wee dog Palsy, her phone and money – why she did that is still to be established.
“We know that earlier in the day she appeared somewhat distressed whilst travelling on a bus, so much so that the driver called police and she was taken home to family, however, for whatever reason, she headed out around 9pm.
“While there is nothing at this time to indicate any suspicious circumstances or that anything untoward has happened to Janet, it is very unusual for someone to just disappear without any trace.”
Police have carried out widespread door-to-door inquiries and a stop-and-interview operation, as well as a family press conference.
Officers have also looked through hours of CCTV and liaised with her bank in case there has been any activity on her account.
Mr Hutton said: “Janet’s family are desperate for her to come home. We are doing everything we can to find her.”
Ms McQueen is described as white, 5ft, slim and with short black hair.