The talks cooled after Schnatter's use of a racially charged slur led to his resignation. Schnatter still sits on Papa John's board and has a nearly 30 percent stake in the company.
Shares of the pizza company jumped after the Wall Street Journal first reported news of the talks, and ended the day up 4 percent at $53.60.
The source requested anonymity because the information is confidential. Both Wendy's and Papa John's told CNBC they do not comment on market rumors or speculation. Representatives for Schnatter declined to comment.
The deal talks highlight the opportunity potential acquirers see in improving Papa John's technology and scaling up in a still fragmented pizza chain industry. Still, the public relations fall out that followed Schnatter's resignation is likely to complicate any potential deal.
His resignation as chairman came after he confirmed that he had used the N-word during a May conference call with media agency Laundry Service. Since then, Papa John's has worked to distance itself from Schnatter, prohibiting him from talking to the press, removing him from the pizza chain's advertising materials and revoking his office space at the company's headquarters.
Still, analysts at Stifel recently wrote, "we believe whoever owns the company will need to fully rebrand Papa John's, requiring more than just advertising and packaging changes to alter consumers' perception."
"A critical factor for the company or a potential buyer is determining the level of investment and duration needed to stabilize sales and profitability and return the company onto a growth path," they added.
The pizza chain earlier this week formed a special committee to "take action with respect to all of the company’s relationships and arrangements with John H. Schnatter," sparking takeover speculation.
As for Schnatter, he doubled down this week on claims he made during a television interview Friday. Schnatter alleges Laundry Service had provoked him into using the term and then tried to blackmail Papa John's for $6 million to keep quiet about it.
Laundry Service disputed his comments in an internal memo sent to employees Tuesday, calling his claims "disparaging and outrageous."
Los Angeles-based attorney Patricia Glaser, who's representing Schnatter, said she asked Laundry Service for a recording of the call, if one exists.
“If there’s a tape, let’s hear it,” Glaser told CNBC. “We believe it will vindicate John, and we want to know if it exists."