The Uruguay international has reacted to the sniping by producing the form of his life in the past month, soaring to the top of the Premier League scoring charts with some audacious, and dazzling displays of the attacking arts.
For Rodgers, that is confirmation the striker is an inspiration to his young side.
"Your number nine is always a reference for the team, whether it is through goals or his presence," Rodgers said.
"With Luis, he's a real reference for this team, because of what he brings - what you get is an all round game. He has great confidence, and yes, that transmits itself to the rest of the team.
"We want to work and press the ball high up the field and if you have a number nine who doesn't want to do that, that can be difficult for you. You cannot function as a team that way.
"But Luis offers a great presence up there, and is a great reference for this team, which is important to us."