The alleged couple became romantically involved while working on the television movie "Northern Lights," said the magazine, which also published images from a video security camera that appear to show the co-stars holding hands and kissing earlier this month.
Rimes, 26, has been married to former backup dancer Dean Sheremet for seven years.
Cibrian, a 35-year-old actor who has appeared on "Ugly Betty," "Invasion" and Samantha Who?," has been married to model Brandi Glanville since 2001. Usmagazine.com said they have a 5-year-old son and a 23-month-old son.
"This is a difficult time for me and my loved ones, but I appreciate all your continued support," Rimes said in a message posted on her Web site Wednesday. "I would like to assure all of you that this is a place for you to hear things directly from me and, as you all know, not everything in our lives is always black and white. Have faith, Le."
Us Weekly said Cibrian's manager didn't respond to its request for a comment on its story.
However, the actor issued a statement denying the report to TV's "Access Hollywood" program Wednesday.
"Other than being friends and two actors who were romantically intertwined in a movie-for-television we filmed last fall, there is no truth to the reporting initiated by US Magazine," Cibrian said.
"Northern Lights" is slated to air Saturday on Lifetime.
Notable deaths of 2014 [PHOTOS]