French newspaper Midi-Olympique claim the Parisian club are to make the 25-year-old a lucrative offer to quit Glasgow Warriors when his current contract expires at the end of the season.
Russell, who was called up as cover during the British & Irish Lions tour of New Zealand this summer, has 29 caps for Scotland.
As one of Scottish rugby's star men, it would be a blow to the Scottish Rugby Union if he were to be tempted overseas.
Racing already have All Blacks legend Dan Carter on board but the veteran fly-half is also out of contract at the end of the season.
He is, however, believed to be weighing up the chance to stay with the Top 14 club for another year.