While Cox has forecast only "little tweaks" to his predecessor John Longmire's coaching systems, he says enticing Warner to sign a fresh contract is paramount.
The West Australian-born midfielder is already understood to have been courted by Perth clubs ahead of coming out of contract at the end of 2025.
"For us, it's about trying to make sure that Chad is as happy as he possibly can be, which he is," Cox told SEN WA radio on Wednesday.
"We'll speak to Chad throughout the process to make sure that hopefully (we) can get it done as soon as possible.
"Chad is at the top of our priority now.
"He's an amazing footballer, one I've got a really strong relationship with, he is really enjoying his time in Sydney.
"That's the role of our footy club and myself now, to try and make sure we can get Chad locked away as quickly as possible and create the environment he wants to be a part of going forward."
Longmire officially stood down as the Swans' senior coach on Tuesday after 14 seasons, 333 games, five AFL grand-final appearances and one premiership.
The Swans had routinely denied a succession plan had been in place to replace Longmire with his long-time assistant Cox, who was only offered a four-year deal last Tuesday.
But Cox had always been privy to Longmire's plans to resign at the end of either the 2024 or 2025 season - aspirations held by Longmire for 18 months but only revealed publicly at Tuesday's press conference.
"We've sort of stayed in constant dialogue all the way through. The possibility arose and I'm really excited about it," Cox said on Tuesday.
An assistant to Longmire since 2017, Cox had long been tipped as a senior coach in waiting and most recently repelled advances from former club West Coast this season.
The 2006 premiership player was the Eagles' ruck coach for three years after his retirement in 2014.
(L-R) Swans chairman Andrew Pridham, John Longmire and Dean Cox arrive for the press conference. (Dan Himbrechts/AAP PHOTOS)
Cox said he would not have quit that job and relocated his young family to Sydney had he not felt the move would one day lead to a senior coaching opportunity.
The Swans' transition to Cox comes 14 years after Longmire replaced outgoing premiership coach Paul Roos following his own apprenticeship as an assistant.
Chairman Andrew Pridham joked at Tuesday's press conference he would see Cox "in 14 years" - eluding to another regeneration from within the club.
A big turnout for a big moment in our club's history. — Sydney Swans (@sydneyswans) Watch live: https://t.co/tTAfxkYmgg pic.twitter.com/ItgOXguGJuNovember 26, 2024
The Swans have remained almost perennially successful since Roos took charge in 2002, making finals in 19 of 23 seasons.
Cox paid homage to that stability in eluding to his plans for next season.
"One thing that I'm extremely proud of this organisation, is the sustainability of performing to the highest level," he said.
"Our players produced some amazing football this year and that's something that we're extremely proud of. There'll be some little tweaks that I do differently from John but overall, it's a well-run footy program.
"It has been for a long time and we'll make sure that we give the players the best chance that they can continually perform."
Dean Cox is chaired off the ground after playing his last Eagles home game before retiring in 2014. (Theron Kirkman/AAP PHOTOS)
The six-time All-Australian played 290 games for West Coast, including the 2006 premiership triumph over the Swans.
"My playing career ended and when it finished I was all about how can I try and help players experience what I experienced?" Cox said.
"For me, it's about trying to make sure that they have an experience that they play some really good footy, they enjoy doing it together and then when they leave, hopefully their life is set up that they can go onto the next phase."
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