On July 1, the 28-year-old failed to show up to work at a school in Kanagawa, a prefecture south of Tokyo, where she works as an English teacher.
Police have launched a search operation and Ms Hodgkinson’s family have appealed for information on social media.
Ms Hodgkinson’s brother, Peter, said: "We have Nottinghamshire Police setting up a link with the police in Japan over Interpol to deliver news and information to us, but this does take a few days to get setup.
“The Foreign, Commonwealth and Development Office also have an incident logged."
He added: "I want people to be found that are missing, I want everyone to feel safe and confident living their life to the full.
"Whilst focusing on Alice I didn’t expect the impact that this would have on so many people’s lives."
According to the Guardian, police reportedly visited Ms Hodgkinson’s Yokohama apartment and broke down the door. They said they had found a note but have declined to reveal what it said or if it had any link to her disappearance.
Her father, Stephen Hodgkinson, told The Telegraph that he spoke to his daughter the day before she disappeared and there was “nothing out of the ordinary at all.”
He told the newspaper: “We would hear from her pretty regularly and catch up on Skype, but the first I heard that something was wrong was when the manager of her school called to say that she had missed work for a few days and that they were concerned about her well-being.
“I last spoke to her on the day before she disappeared and there was no indication that anything was wrong. She had sent me a Father’s Day present and an e-mail and I wanted to thank her.
“There was nothing out of the ordinary at all, nothing suspicious.”
Ms Hodgkinson’s family are being assisted by Nottinghamshire police, Interpol and the British embassy in Tokyo.
A spokesperson for the Foreign, Commonwealth and Development Office said: “We are in contact with local authorities in Japan regarding a missing British woman. Consular staff are in touch with her family in the UK.”
Anyone with information is asked to contact the Kanagawa prefectural police headquarters on (+81) 045 211 1212.