The newspaper reports that the Tampa Bay Buccaneers have also requested permission to interview Dawson for their general manager opening.
Dawson just finished his seventh season in the Titans front office, and second in his role as vice president of player personnel.
Dawson, who was interviewed for the Rams GM job two years ago, works alongside general manager Ruston Webster, managing day-to-day football operations.
Dawson has had three executive positions with the Titans since 2007, and during that timespan the Titans have produced three winning seasons and a 59-53 overall record. The Titans recently fired head coach Mike Munchak after he produced a 7-9 record in his third season.
Dolphins owner Steve Ross is seeking an experienced talent evaluator to replace Jeff Ireland, who held the position for six seasons before being relieved of his duties on Tuesday.
The Dolphins plan to begin the interview process on Friday. Ross is searching for an experienced talent evaluator, and prefers someone with a proven track record for evaluating football talent.
"A football person," was how Dolphins CEO Tom Garfinkel described the ideal candidate on Wednesday. "Intellect, a collaborative team player. Someone who is creative and has an open mind.
"We need someone who works really well with others, and puts the team, organization first."