The celebrity astrologer, who competed in last year's series, has been added to a shortlist drawn up by BBC chiefs following Dixon's defection to Britain's Got Talent earlier this month.
"Russell made a huge impression on Strictly last year - not only with viewers but with bosses too," an insider told The Sun.
"There was a big groundswell in favour of Russell taking over Alesha's judging role, and executives contacted him to see whether he would be up for it and he gave them a resounding yes."
Although Dixon's replacement is expected to be female so as to prevent an all-male panel, the source insisted that anybody was capable of landing the spot next to Len Goodman, Craig Revel Horwood and Bruno Tonioli.
"There are some concerns that Alesha's replacement should be a woman as an all-male panel would be too lop-sided, but executives will decide on that in the spring."
Grant's Strictly co-stars Nancy Dell'Olio and Chelsee Healey have also been linked to the role recently, though pro dancer Karen Hardy has long been established as favourite to replace Dixon.
Watch Russell Grant perform on Strictly Come Dancing below: