During a press conference yesterday, police said Miss Buckley had travelled to the man’s flat in Dorchester Avenue in his car. She was last seen when she left the flat at around 4am and was planning to walk home to her flat in Hill Street.
Police said the man was “helping with inquiries”, but was not being treated as a suspect. Last night, a cordon was in place outside the property.
The family of Miss Buckley, who is from Cork in Ireland, have flown to Scotland to help with the search.
At the press conference in Glasgow, Miss Buckley’s mother Marian made an appeal for anyone with information that could help police to come forward.
She said: “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.”
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.
Miss Buckley was seen on CCTV outside, talking to a man and then walking west along Dumbarton Road towards Church Street.
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.
A handbag which police believe belongs to Miss Buckley was found in
Dawsholm Park, near to Dorchester
Avenue, yesterday afternoon. There has been no activity on her social media accounts in the last couple of days and her mobile phone is believed to be out of battery.
Police are also keen to find out more about a grey car that was seen on the roads between Milngavie and Drymen north of Glasgow between 11am and 3pm on Monday.
Detective Superintendent Jim Kerr said: “We’ve traced the man she was with in Dorchester
Avenue. He believes he was intimate with her at his flat consensually in the early hours of Sunday.”
Mr Kerr said: “The activity of this grey car on the Monday is something that we’re a wee bit concerned about. This car has been seen at various locations on these roads and I want to know why.”
He added: “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.”
Police have also traced a second man to whom the student spoke in the club before she left.
Miss Buckley’s father John, 62, said: “We are extremely concerned for her. We are desperate to get her back and safe with her family. She is our only daughter; we love her dearly and just want her to come home safe and sound.”
Mr Buckley described his daughter as a “sensible girl”.
“She always keeps in contact with her family and friends to let them know where she is and what she is up to,” he said. “To not return to her flat or be in contact with her pals, not responding to texts and calls to her mobile, is very, very worrying as it is so out of character.”
Det Supt Kerr added: “Karen is a lovely girl, close to her family and popular with her friends. She has only been in Glasgow a few months and if staying out will always let her friends know – hence the real concern here.
“Like all youngsters, she is also very active on social media but there has been no activity on any of her accounts since Saturday night/early Sunday.
“This is understandably a heartbreaking time for her family. Of course, there is an intense interest both from the public and the media regarding her disappearance; however, speculation, particularly on social media, about the aspects of the case is not helping our inquiry and is extremely distressing for Karen’s family and friends.”
Miss Buckley is a first-year occupational therapy student at Glasgow Caledonian University. She moved to Glasgow in February and before that was a nurse at the Princess Alexandra Hospital in Harlow, Essex.
She has three older brothers: Brendan, 32, Kieran, 28, and Damian, 27.
Police said that she had travelled widely abroad and was aware of how to take care of herself.
More than 27,000 people have now “liked” a Facebook page set up to help find the missing 24-year-old. Kimberley Smith, Miss Buckley’s ward sister when she worked in Essex, was among those posting messages of support. “Such a lovely girl,” she wrote.
Speaking to the Irish media earlier yesterday, Miss Buckley’s brother Kieran said the family was praying that there was some “crazy excuse” behind his sister’s disappearance.
Mr Buckley said: “We’re trying to stay positive and hope some good news comes.”
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