The Victorian building, in Brunswick Road, is so haunted that some staff members have been left too scared to go into certain parts of the museum.
But museum assistant Nigel Taylor-James, who has worked there since 1989, says George is a friendly soul.
"From what I have seen he has a medieval appearance, with a hooded outfit made from simple cloth," said Nigel.
"I've seen him three times. Each time he looks at you and then makes a hasty exit. He doesn't stay around for long.
"Before I had seen him I was a non-believer. It is one of those things that you have got to see with your own eyes to believe.
"He is like an apparition. I have never been scared at the time. It is not a scary experience, more one of surprise."
There are a number of theories as to who George could be.
The museum was opened in Victorian times but a much longer history can be found on the site.
In Roman times it was at the edge of the city and a Roman skull has since been found.
But the nearby Greyfriars friary church would have had its graveyard nearby. And there is evidence of a Roman cemetery in the vicinity too.
Museum curator David Rice said: "I have been here for 13 years and over that time lots of people have come to me and said they had seen him.
"I have seen something myself and there have been many incidents of doors opening and closing, footsteps and strange noises.
"The museum is filled with thousands of people's belongings from all around the city and it was once a place were human remains from excavations were kept.
"Some people have come to me scared after they had caught a glimpse of him.
"He's usually around in the daytimes but the museum is closed in the evenings so he may well have free rein of the place. He's a friendly ghost anyway."