Elizabeth Brown, or Hunt, was last seen getting off a number three bus on Nicolson Street at 11:55am on Friday, 28 May.
The 55-year-old, known as Betty, was reported missing last Tuesday after failing to turn up for work following a week's holiday.
Originally from Hawick, police believe Mrs Brown may have travelled to her hometown from Nicolson Street but sources say her disappearance is out of character and police are understood to be concerned.
But today Betty's older sister Maureen McLauglin revealed her family was "worried sick" and urged the mother-of-two to contact relatives or police.
"Betty, we want you to know that whatever problems you are having, we can work through them together as a family," she said.
"Everyone misses you so much, and all we want to know is that you are safe and well.
"If you don't want to get in touch with us directly, please contact the police or find some way of letting us know that you are OK. We are all worried sick, and desperate for any news."
Police are to renew their appeal for information today by handing out leaflets and speaking to passengers on the bus route where Betty was last seen.