The 20-year-old is off-contract and toured the Dolphins facility recently when the Wayne Bennett-coached outfit made their pitch to the Ipswich State High product.
Bennett and Dolphins CEO Terry Reader have both said the club has plenty of room in the salary cap and funds tucked away for the right players.
Finefeuiaki has been identified as a quality footballer and a youthful one to add to a veteran Dolphins pack.
Wayne Bennett (centre) says the Dolphins have enough salary-cap room to fit in Finefeuiaki. (Darren England/AAP PHOTOS)
"He is an explosive young player on the way up, with a lot of ability," Bennett said.
"We have just got to wait and see now what he does."
The Cowboys don't intend to let the youngster go.
"The kid is still learning, but the upside is broad and we are hopeful and optimistic that we can keep him," Cowboys coach Todd Payten told AAP.
Todd Payten is confident the Cowboys can keep hold of rising star Kulikefu Finefeuiaki. (Scott Radford-Chisholm/AAP PHOTOS)
"He doesn't mind confrontation. He's got great leg speed and he runs hard and tackles hard."
Finefeuiaki is starting for the Cowboys due to a syndesmosis injury to fellow back-rower Heilum Luki, who is also off-contract.
"Kuli's a fine young man who is certainly learning his craft and quite explosive in what he does," Cowboys CEO Jeff Reibel said when asked about a new deal for him.
"We are also speaking with Kuli. We are quite confident that those talks are remaining positive, and when we have the opportunity to update we will do so.
"I think salary-cap management is an art, rather than a science as these guys come on.
"Being north Queenslanders, we want to look after and nurture our own.
"From my perspective ... Kuli is a priority, as is Heilum to ensure they have long-term futures at our club."
Cowboys 2015 premiership-winning second-rower Gavin Cooper also hopes to watch Finefeuiaki ply his trade with the club long-term.
Cowboys 2015 premiership player Gavin Cooper is a big fan of Finefeuiaki. (Darren England/AAP PHOTOS)
"He's a big boy ... a power athlete," Cooper said.
"He still plays very raw. Sometimes, respectfully to him, he's like a deer in the headlights when there's a lot happening around him, but you give him the ball and you've got to move out of the way.
"I hope to see him in that Cowboys jersey for a long time.
"I know there'll be some teams throwing some dollars at him, but I think he knows what's on the horizon for the Cowboys and I hope to see him stay there for a little while.''
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