The Russian has been linked with the Toffees as he comes off contract with the Norwegians.
Khaykin's first senior goalkeeper coach Dmitry Gulenkov told Sport24: "For three years in Norway, Nikita has grown enough in terms of level.
"Play in England? Rumours are always rumours. He has a good chance to play at Everton. Nikita is not the first season shows himself from a good side.
"Of course, there are all prospects to play in England. Everything is possible!"