The Las Vegas Review-Journal reported Wednesday that the governor said he has been a "friend" of the woman for more than 15 years but "there is nothing inappropriate about that friendship."
Gibbons has filed for divorce from his wife, Dawn, of nearly 22 years. He has requested that the divorce proceedings be sealed.
Nevada law allows one party in a divorce to close the case records and request that the divorce be conducted out of public view.
In a May 21 editorial supporting Dawn Gibbons' efforts to unseal the divorce case, the Elko Free Press said the governor was "involved" with the woman, but did not cite sources for its allegation.
In a letter to the Elko newspaper, Gibbons said the editorial was "filled with innuendo and quite frankly, tabloid-style gossip. It was absolutely out of bounds of good taste and common decency."