Wales' double Grand Slam and RBS 6 Nations title-winning mastermind will be interviewed for the post along with Ireland boss Declan Kidney and Scotland's Andy Robinson, Press Association Sport understands.
But it is thought that New Zealander Gatland's appointment is viewed as a formality for the three-Test trip to Australia. It should be confirmed next month.
"I love working with him - he is a fantastic coach," Warburton said.
"If you look at his record - two Grand Slams and a World Cup semi-final - he deserves the Lions job.
"He gets the best out of players, and I owe a lot to him. He put his faith in me as a young player and captain."
Warburton, meanwhile, is due to see a specialist in an attempt to gauge the extent of a shoulder injury that forced him off during last Saturday's RBS 6 Nations title clincher against France.
It was his third different injury of this season's Six Nations following earlier thigh and knee trouble that contributed to him playing just 160 minutes from a possible 400 in the championship.
Warburton has already undergone an MRI scan, but that proved inconclusive.
"Not much showed up on it," he added. "There is a bit of concern because I am still in quite a bit of discomfort.
"I am seeing a specialist, and I will find out more then. The MRI scan did not show any soft issue damage.
"It may be that it is more complicated than that, such as something to do with nerves, which would not have shown up.
"I do not know what to think - it could be two weeks, or it could be two months. I hope it is shorter, rather than longer.
"One of the reasons they are finding it difficult is that I cannot recall the incident in the game when it happened.
"If the physios knew when it happened then they could look at the video and see what sort of collision it was and what might have happened to the joint.
"There was a lineout when the ball was overthrown, and Julien Bonnaire beat me to the ball. I went to grab it, but I could not lift up my arm because of the pain.
"I knew then I would have go off at half-time. I have not watched the game back yet, and I will have to do so to see if it triggers something in my memory."
Warburton, though, has confirmed it is not the same shoulder that he underwent reconstruction surgery on three years ago.