Karanka - who worked with the player while he was Jose Mourinho's assistant at the Santiago Bernabeu - says he hopes to convince Jese to try his luck in England on what is expected to be a loan deal.
"Jese Rodriguez is a player I have worked with and I know how good he is. His next move is important and I hope we can convince him," Karanka told a news conference.
"He was playing for Real Madrid last season. He's a Champions League player, a young player with an amazing future in front of him."
Karanka, who confirmed defender Calum Chambers has suffered a stress fracture in his foot, also said Boro have not given up hope of signing Robert Snodgrass from Hull City.
West Ham are also said to be keen but Karanka says his side are still in the hunt.
"He's a player I like. The club is working on offers with Hull. He's a player I like a lot," he said.
"I think they rejected an offer from us but the club is working. He's a player who we like and who we want to bring here."
Karanka could be given a financial boost following Gaston Ramirez's decision to submit a transfer request.
The 26-year-old is a reported target for champions Leicester City but Karanka warned that he will not be sold on the cheap.
"Yes, he's handed in a transfer request. It's frustrating," he said.
"He was calling us to play here last season, he got back to his best here, he got back in the [Uruguay] national team. It's frustrating as a coach and a person.
"Nobody is bigger than the team. At the end, if we don't receive the right offer, he will be an important player."