The 31-year-old Argentina international has been heavily linked with a return to his homeland, but the former Swans boss wants to extend the winger's stay at Anfield
Rodriguez, 31, struggled to force his way into Kenny Dalglish's first-team plans for most of last season, having to settle for a bit-part role before then being told that he was surplus to requirements.
However, Rodgers believes Rodriguez has a future at the club and is eager to convince him to stay despite rumoured interest from former club Newell's Old Boys.
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"I have known about Maxi for a long time. He is experienced, a very good player and he will understand 120 per cent about the way I want us to play," Rodgers told the Mirror.
"I would like him to stay, but sometimes there are issues that surround that and I understand that - which is why I am speaking to him now.
"But I would love to have him here, I think he is a very good player. Whether it was starting or coming off the bench, he would be very good for our squad, because he understands what we want.
"He doesn't speak the language, it has been difficult for his family, so if a player like him wanted to leave, it would not be because of the club.
"I have had a couple of real good conversations with him. I know his thoughts and how he is thinking.
"It was a difficult season for him and he probably felt he should have played a lot more. We will hopefully see where he is at over the next couple of weeks and find a solution."
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The 39-year-old boss was also quick to lend his backing to fellow midfielder Jordan Henderson who endured an indifferent first season at Anfield after his big-money move from Sunderland, alluding to a key role in his new-look Liverpool side.
"Jordan got a wee bit of stick last year but do you know what, this is a boy who is 21 years of age," Rodgers said.
"He's come here for a bucket load of money, to a massive club, with massive expectation. He's moved away from home, he's on his own and he has to be a world beater straight away.
"It was never going to happen. When I first saw him at Sunderland I thought 'this kid is going to be a really good player' - and this kid can play football.
"If he is in a certain system - in an environment which is going to help him and educate him in the game - you will see he's a player."