Kate Hunter, from Wetherby, was reported missing at 12.25am in the early hours of Saturday after she failed to return home on Friday night from Leeds city centre.
Her sister has appealed for information on Facebook, saying her family are “desperate” to find the teenager.
West Yorkshire Police checks on CCTV in the city centre have traced her last known movements to McDonald’s, in Briggate, where she was between 10.16pm and 10.42pm on Friday.
Officers have now released CCTV images of an unidentified man who was seen with her that they urgently want to identify.
Detective Inspector Marc Bowes, of Leeds District Safeguarding Unit, said: “Given Kate’s age and the length of time she has now been missing, we remain very concerned for her welfare and need to establish her whereabouts as soon as possible.
“The CCTV images from McDonald’s show her in the company of an unidentified man and we urgently need to identify him.
“There could be any number of explanations as to the link between Kate and this man and we are keen to avoid unhelpful speculation at this stage.
“We are not seeking his identity as suspect for any offence but want to speak to him as someone who is likely to have information that could assist in tracing Kate. We would like this man or anyone who recognises him to contact us immediately.
“Kate’s family are understandably worried about her and need to know that she is safe and well and we would urge anyone who has any information that could assist in tracing her to get in touch.”
Anyone with information is asked to contact the Special Operations Room at Elland Road via 101 quoting log number 47 of December 17.
In a message on Facebook yesterday, Kate’s sister Natalie Horner wrote: “My beautiful sister Kate is STILL missing. It has been almost 48 hours and we are no closer to finding her, please please if you have any information call us or message us, we are totally desperate.”