The 16-year-old is said to have agreed a two-year youth deal with the Trotters.
He is a shot-stopper of considerable promise and is being backed to emulate the success enjoyed by fellow Hungarians Adam Bogdan and Marton Fulop in English football.
Daniel Illyes, who worked with Bukran at the Eger Goalkeeping Academy, told Nemzeti Sport: "He's an enormous talent, there is no question of that.
"Erik signed to us at the age of 13 and we saw he had a big future. A year later Budapest Honved signed him and now we can say that Bolton have signed him for two years on a fellowship contract.
"It is only two years because the FA only allows professional contracts from the age of 18. Honved have done good business with Bukran's transfer.
"It is evident that besides his talent, Adam Bogdan helped to take him to Bolton. They thought, 'If a Hungarian was a good choice before then why shouldn't we sign another one?'"