The Liverpool Echo says the Glasgow giants are believed to be interested in taking the 21-year-old striker to Parkhead for the new season.
Moyes has not given up on Vaughan despite his struggles with injuries, including a recent 19 months on the sidelines.
Vaughan was allowed to join Leicester City for a loan spell last season, but continued to suffer because of injuries and the move north of the border would be a fresh start in a league where fellow Everton youngster Lucas Jutkiewicz has also flourished.
Vaughan, who became the youngest player to score in the Premier League when he netted against Crystal Palace in April 2005 aged 16, surpassing James Milner, has fallen down the pecking order at Goodison since the summer signings of Joao Silva and Jermaine Beckford. But Moyes is keen to give him every chance of fulfilling his early promise and could see the move to Scotland as a way of kick-starting his stuttering career.
Celtic would be able to cover his wages should they attempt to take him to Glasgow to boost their strikeforce and Moyes may be keen to do his old club a favour by sanctioning a deal.