The (Columbia, S.C.) State and The Kansas City (Mo.) Star reported sources said Martin will be formally introduced Tuesday as the Gamecocks' coach.
Martin, 46, has a 117-54 record in five seasons with the Wildcats.
University of South Carolina trustees said Monday they would take up a "contractual matter" Tuesday. The special session will be held via teleconference, The (Charleston) Post and Courier reported.
Gamecocks Athletic Director Eric Hyman is looking to replace Darrin Horn, who was fired March 13 after four years at the school, including three straight losing seasons.
Sporting News reported during the weekend that South Carolina had asked Kansas State for permission to talk to Martin, who guided the Wildcats to the NCAA Tournament four times.
"We have had conversations," Richard Katz, Martin's agent, told the Star.
Martin is under contract with Kansas State through 2015 at a salary of $1.55 million per year, on average, with a $1 million buyout.
Horn made $1.1 million a year and South Carolina owes him a $2.4 million buyout, The Post and Courier said.
Wichita State Coach Gregg Marshall had been considered a likely candidate at South Carolina, but it was announced Saturday he would stay at State at a higher salary.