The series stars James Wolk as a Texas man living a double life with two different families. Adrianne Palicki and Jon Voight also appear in the programme.
The show's premiere was broadcast on Monday night but The Hollywood Reporter says that it received average ratings of just 4.1m viewers despite airing after House and receiving good reviews from critics.
Some reports have suggested that the show struggled to compete against Dancing With The Stars, Two and a Half Men and NBC's new series The Event.
Rumours have now claimed that Fox is considering axing the show, temporarily pulling it from the schedules or finding a different timeslot for it.
One agent, who represents a Lone Star writer, said: "No-one in TV should be happy about this. This is going to have a chilling effect on networks taking chances on anything but cookie-cutter shows."
Fox declined to comment on the rumours.