Papua New Guinea captain Martin, 28, joined Rhinos midway through the 2019 season from NRL side Canterbury Bulldogs and has now made 45 appearances for the Rhinos, scoring 372 points for the club.
Martin said: “I am pleased to have got my future agreed.
“It took a bit longer than we all wanted but I am very happy to have got it sorted now and I’m looking forward to finishing this year without that on my mind and finishing the season strongly.
“There is a good spirit in the camp at the moment ahead of a big game against Hull FC this Saturday at the Magic Weekend. “It feels like the play-offs are starting early.
“They are in the same boat as us whereby every win is crucial. “We have to focus on our performance and hopefully it will be one of the games of the season for us and we can get the win.”
Rhinos head coach Richard Agar said: “I am pleased for Rhyse that he has agreed a new deal.
“I know he has felt at home here at the Rhinos and is an integral part of our squad.
“His goal-kicking has been a big asset to our team this season and he is a popular member of our group.
“He will be a big part of our squad next season as we look to build strength in our pack in particular.”
Tonga centre Hurrell was signed from Gold Coast Titans in 2019, and has been linked with a move to St Helens while Fijian prop Vuniyayawa joined ahead of this season after leaving New Zealand Warriors.
Agar added: “It is disappointing to see Konrad and King leave the club but that is the nature of a salary cap sport.
“They have been integral members of our squad this season and, in Konrad’s case, for a number of years.
“Konrad has been a fantastic player for Leeds Rhinos since he moved to Emerald Headingley.
“He has been a great team-mate and personality and I know he has a wonderful rapport with our fans because of the passion he always brings to the Leeds Rhinos shirt.
“When King joined us we took a chance on him but he took a chance on us too.
“It is a credit to him and his work this year that he has now received multiple offers for his future.
“We fully understand he is looking after his longer term future with his next contract and, whilst we are sad to see him go, I am pleased for him.
“They will both have a big part to play this season within our group and will leave with our best wishes and thanks at the end of the campaign.”