"Sheryl and I were in the car yesterday when somebody called us about the story," he told TV Guide in a telephone interview from Austin, Texas. "It is completely untrue."
Armstrong and the Grammy-winning singer met in 2003, the same year that he and his first wife were divorced. He refused to say if he and Crow are planning to marry.
Winning six consecutive Tours after surviving cancer has made Armstrong familiar with rumors, usually that his speed and stamina have been boosted by performance-enhancing drugs.
Armstrong plans to retire once he reaches the finish line in Paris in July. The Tour begins July 2.
Notable deaths of 2014 [PHOTOS]