Police searching for missing Irish student Karen Buckley have said they are keen to trace a grey or silver car seen in the area on the night she vanished.
At a police conference this afternoon, officers said they were "extremely concerned" for Karen's welfare.
But they stressed the man, seen with her on CCTV shortly before her disappearance, is not a suspect in the case.
He drove Karen to his apartment on Dorchester Avenue in Glasgow in his car after the pair left the Sanctuary nightclub at approximately 1am on Sunday morning.
According to the man – who is now helping the police with their enquiries – Karen spent some time with him in his flat.
Karen is thought to have left his residence after 4am and walked home, said Detective Superintendent Jim Kerr.
"We believe she got into a vehicle near church street and went to another location, leaving there after 4am."
"She spent some time with the man from the CCTV footage."
The investigation is now focused on this grey/silver car which was seen in the area at the time.
“It was spotted on country roads near Milngavie. We’ve no make or model number for the vehicle but we are trying to trace it. We want to know what it was doing in the area.”
Stressing that they were still viewing the investigation as a ‘missing person’ case, officers said they were ‘gravely concerned for the safety of Karen.’
They also revealed that 20 minutes before the press conference commenced, a member of the public handed in what is understood to be Karen's handbag.
The bag was found in Dewsholn Park, close to Dorchester Avenue where the man lives. Dewsholn Park is about two miles from Sanctuary nightclub.
Ms Buckley's mother Marian and father John were also at the press conference.
Her mother said:
"We just want Karen home safely. We are desperate, she is our only daughter. We love her dearly and if anybody has any information please come forward. We would dearly appreciate it. Thank you."
Her father, John said:
"Like Marian said, if anybody is holding Karen or anybody knows anything about her, just please return her please. Just give her back, we love her to bits. We're devastated. Thank you."
Earlier today, her brother Kieran said the family are praying that there is some “crazy excuse” behind the Cork student’s disappearance, adding that “all we can do now is hope she turns up safe and well.”
Speaking from Sydney, Kieran (28) said the family were doing all they could to “stay positive” since the 24-year-old nursing student vanished after leaving a popular Glasgow nightclub on Sunday morning.
“We’re trying to stay positive and hope some good news comes,” he said.
“We’re just hopeful she will turn up safe and sound… [and] that there’s a crazy excuse for Karen’s disappearance. All of this is completely out of character for her.”
Speaking on Newstalk Lunchtime, Mr Buckley said he was boarding a flight to Scotland to help support his parents and aid in the search for his sister.
“I’ll do whatever I can to – search with the police, be there to support my parents. It’s hard for them, for all of us, so I’m just going to be as helpful as possible.”
“I’m over here with my brother and its been very tough for both of us because it is hard to get information with the time barrier.”
“We found out on Monday morning that she had gone missing and we just hoped she would turn up - that maybe her phone had just gone died.”
“But then the police got involved and now we’re just tying to get to the bottom of this.”
Earlier today, Scottish police confirmed they had traced the man seen talking to Karen outside The Sanctuary nightclub in Glasgow in the early hours of Sunday morning.
The individual is believed to be helping with their inquiries, and is thought to have been the last person to see the Cork native before she disappeared.
The man was seen on CCTV talking to the young student outside the nightclub and, until now, had been the main focus of inquiry since her disappearance.
Speaking to Independent.ie, Greater Glasgow police said they were still continuing an intensive search for the missing Irish student.
Asked about earlier reports that they were searching for a second man, a spokesperson for the force said this was not the case, adding that they could not comment on rumours that Karen was active on Viber, an online messaging app, hours after she is thought to have disappeared.
Friends of the nursing student however told Independent.ie that they had had no contact from the 24-year-old since she was last seen talking to a man outside the Sanctuary night club before walking westwards along Dumbarton Road away from the city centre towards Church Street.
The family of Karen, who disappeared after telling friends she was going to the toilet in the nightclub, have flown to Scotland as the search for her continues.
They are due to taken part in a police press conference later today at 2pm in Glasgow, where it is believed they will issue an appeal for help in tracing Karen.
Ms Buckley's worried parents John and Marion travelled from the family farm Mourneabbey near Mallow in Co Cork to Glasgow yesterday, and are thought to be leading efforts to find their only daughter.
The youngest of the family, Karen has three older brothers - Kieran, Damian and Brendan.
Speaking to Independent.ie yesterday, brother Kieran said: "It is so hard. I am in Australia and feel helpless."
"We're trying to do what we can from here... We just hope she is found safe and well.
"She doesn't just go away or go off on her own. She would let people know she was coming home, which is why her friends and us are so worried."
Karen is studying occupational therapy at Glasgow Caledonian University having previously studied nursing in Ireland.
She moved to Glasgow in January after working for a year in Essex.
The 24-year-old was last seen on CCTV in the early hours of Sunday morning, talking to a man outside the Sanctuary in Dumbarton Road, Glasgow.
She arrived at the club with friends at around 11.45pm on Saturday and at around 1am told them she was going to the toilet but failed to return and did not take her jacket.
In a statement yesterday, Scottish Police said the student had been drinking in an apartment with friends before leaving to go to the nightclub. The group arrived at Sanctuary nightclub around 11.45pm.
At around 1am she told friends she was going to the toilet but did not return and did not take her jacket.
Karen's father, John, last night said he was unable to comment on the ongoing efforts to find his daughter.
A neighbour described Karen as a "lovely, quiet girl".
"She comes from a very well-respected farming family who would be very well-known in the area," she said.
The Mourneabbey parish is to hold a special mass Wednesday night in Analeentha Church at 8pm for the missing student.
The missing girl's friends have also issued appeals and have set up a Facebook account to help trace her.
Friends of the Glasgow Caledonian University student raised the alarm yesterday, saying it was "very out of character" for her not to return home.
Searches yesterday concentrated on the area in which Karen went missing, including a park, bushes and some areas of the River Kelvin. The search is set to be expanded today.
Inspector Gavin Smith said, "Her friends say she would always contact them by text or phone to let them know where she was and she doesn't appear to have contacted anyone.
"We are interested in speaking to anyone who may have seen Karen in the early hours of Sunday morning in the west end of Glasgow in order to help with our inquiries to locate her."
"I would urge anyone who may have seen Karen at all on Saturday night into Sunday morning to get in touch with police on 101 with any information at all."