Samreen Ahmed, 15, was last seen at her home in High Wycombe on the afternoon of Wednesday, January 2, when she said she was going to the town centre to meet friends.
Thames Valley Police said she had recently been seen with a 20-year-old man, Muhammed Shoyab Khalifa, and may still be with him.
The pair were spotted boarding a train to Birmingham Moor Street and were last seen in Circus Drive, Glasgow, on January 10.
Samreen is described as 5ft tall, of slim build, with very long dyed brown hair, brown eyes and she may be wearing glasses. She has two studs in each ear and a stud in the upper left ear.
Police said there was growing concern about her welfare and have appealed for help from the public to trace her.
Detective Seargent Sally Spencer, who is leading the search said: "She has been missing for 10 days now and even with the help of friends she must be running out of funds and means of support."