Shields took back WBC and WBA belts with a ruthless sixth-round stoppage of Kaye Scott as the unbeaten Michigan fighter became a middleweight world champion for a third time.
Price, the unified welterweight champion, could be the next opponent for Shields and the American's promoter Dmitriy Salita is keen to finalise terms for a battle between two Olympic gold medallists.
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"We're working on it as we speak," Salita told Sky Sports.
"Our intention is to make the fight. Based on my understanding, we have agreed to all the requests from [Price's promotional team] BOXXER and Lauren's side.
"I invited Lauren and her team to the fight because my relationship with the fighters we promote, and certainly with Claressa, is that they know all the moving parts and are business partners in making these decisions.
"I wanted Lauren and Claressa to have an opportunity to sit down together and see if there was anything outstanding. Claressa is extremely intelligent and very experienced in making big fights, and I'm sure Lauren is sharp as well.
"Unfortunately, they weren't able to make it, but the conversations are ongoing and we want to make the fight."
Shields has also been linked with a fight against WBO and IBF world middleweight champion Tamm Thibeault and fellow American Mikaela Mayer, who faces Chantelle Cameron in a unified world super-welterweight title fight in Birmingham on August 29, live on Sky Sports.
"Claressa wants to fight the best and be part of the biggest, most impactful events - that's always been clear," said Salita.
"There are a few other conversations taking place in case the Lauren Price fight doesn't come together.
"We'll make a decision soon, but whatever is next, the goal is for it to be another legacy-making moment for Claressa and for boxing."
Shields attracted a huge crowd to State Farm Arena in Atlanta, including some of the biggest names in boxing and music, and Salita believes his fighter's reputation has transcended sport.
"It was incredible. We were getting calls and requests from some of the biggest stars in hip-hop, music, boxing and culture right up until the final bell," he said.
"Usher, Floyd Mayweather, Kevin Gates, Terence Crawford - and the list goes on. The atmosphere was electric.
"It was a special night in a packed arena and a real reflection of Claressa's star power and how far her reach has grown beyond boxing."
