The parents (below) of the 24-year-old Cork woman have travelled to the city to assist with the search operation.
In a press conference today, they appealed to the public for help in finding her, saying they are "desperate" for her to come home safely.
Her mother Marian told a press conference in Glasgow: "We just want Karen home safely, we are desperate. She is our only daughter, we love her dearly.
"If anybody has any information please come forward, we would dearly appreciate it."
Police confirmed they had found a handbag belonging to Buckley in Dawsholm Park, near Dorchester Avenue, at approximately 2pm this afternoon.
Detective Superintendent Jim Kerr, from Police Scotland's major investigations teams, said police are gravely concerned for Buckley's welfare.
A search was launched for the Irish woman after she was seen on CCTV footage in the early hours of Sunday morning talking to a man outside the Sanctuary in Dumbarton Road, Glasgow.
Miss Buckley arrived at the club with friends at around 11.45pm on Saturday and at around 1am she told them she was going to the toilet. She failed to return and did not take her jacket.
She was seen outside on CCTV talking to a man and then walking west along Dumbarton Road towards Church Street. According to the man – who is now helping the police with their enquiries – they "had consensual sex" in his flat. Police say the man is not being considered a suspect.
Friends of the Glasgow Caledonian University student raised the alarm on Sunday, saying it was ''very out of character'' for her not to return home.
Police believe she may have gotten into a grey car and travelled to Dorchester Avenue before leaving at 4am. They said they were searching for the grey car.
"The activity of this grey car on the Monday is something that we’re a wee bit concerned about. It’s been seen in various locations," Detective Superintendent Kerr said.
"We are gravely concerned that Karen has come to some harm, whether that is down to foul play, criminality or she has taken unwell or had an accident is obviously still to be established."
Miss Buckley is described as white and around 5ft to 5ft 2in. She has brown eyes and dark hair which had long black curly extensions in it. When last seen, she was wearing a black jumpsuit with red high-heeled shoes and was carrying a black handbag. She speaks with an Irish accent.
She lives with three other student friends in a flat in Hill Street, Garnethill, Glasgow, and earlier on Saturday evening had drinks with friends in the flat before going to the nightclub, although her friends told police she was not drunk.
Mr Kerr appealed for people to come forward with any information about the missing woman's whereabouts, saying while it "may seem insignificant to them, however to us it may be the piece of information that leads us to finding her".
He said the public response so far had been "encouraging", adding: "People really do want to help.
"However, at this time we need the information about her whereabouts and her movements after leaving the club."