Turner is beloved by many of his players but also is known for using loud, insulting and profane language with the players.
The NFL is aware of Turner's tactics, which are not unusual for coaches around the league, and wants to speak to him about the culture that existed around Miami's offensive line, sources told ESPN.
Sources inside and outside of the Dolphins' organization said Turner, who has a military background, would "insult, bully and pile it on" Martin, who left the team last week.
Martin spent nearly seven hours Friday meeting with special investigator Ted Wells, who is leading the league's investigation.
Martin has alleged that he was subjected to bullying and harassment by fellow offensive lineman Richie Incognito, who has been suspended indefinitely by the Dolphins for conduct detrimental to the team.
Incognito, who has acknowledged using racist and profane language toward Martin, filed a grievance Thursday against the Dolphins, claiming that his conduct was part of the normal locker-room environment.
Incognito's expedited hearing on the grievance is expected to take place later this week, sources told ESPN. The arbitrator of the hearing wants access to the material that Martin turned over to Wells on Friday, according to a source. Martin did not invite any representatives from the NFL Players Association to the meeting.
The NFL has received multiple tips of other potential instances of harassment inside the Dolphins' organization, league sources told ESPN.
Because so much new information about the Dolphins has flowed into the league office and into Wells' possession, the NFL believes the investigation will take longer than most expected, and it will be "weeks" before the public learns of the results, according to a source.
Sources claimed that the investigation is expected to go well beyond Incognito and his fellow offensive linemen, spanning to the coaching staff and front office.
Incognito had been a member of the Dolphins' leadership council and was a popular player in the locker room. The troubled offensive lineman was forced to fill a leadership void left by the free-agent departure of longtime left tackle Jake Long, who signed with the St. Louis Rams this past offseason.
Long's departure moved Martin into the vacant left tackle spot, playing next to Incognito. Sources told ESPN that Long was the voice Incognito respected the most, and it was Long's empty seat on the leadership council that allowed Incognito to fill that void.
Information from The Associated Press was used in this report.