Johanna Gray, 43, also known as Jo, was reported missing from Gartnavel Hospital in Glasgow on Thursday, April 1.
She left the hospital at around 3.30pm and was last seen walking to Great Western Road in the direction of Anniesland Cross.
Police say she may still be in the Glasgow area but she had reportedly mentioned visiting the Isle of Bute.
Ms Gray is originally from Dunfermline but lives in the Yoker area now.
She is described as a white woman, 5ft 3in tall, of average build, with blue eyes and brown hair.
She was last seen wearing blue jeans, white shoes and a light-coloured long-sleeved top.
She may also be wearing glasses.
Inspector Gavin Smith, from Partick Police Station, said: "Johanna left hospital unexpectedly on Thursday, April 1, and it’s not clear where she may have gone. She has not returned home and is likely to travel by bus or train.
"As time passes, her family and friends are growing more concerned and we urgently need to find her to make sure she is safe and well.”
Anyone with information should contact police via 101, quoting incident number 2150 of 1 April.