Burke told the Tribune-Review the Penguins and Malkin had been trying to come to terms on a new contract that would keep the 35-year-old in Pittsburgh perhaps for the rest of his playing career, but they so far have been unable to do so.
"We made an offer that we were comfortable with," Burke said.
He declined to give specifics, though Burke denied media reports that the Penguins weren't willing to offer Malkin a four-year contract.
"There are stories out there that we never offered a four-year deal; that's completely false," he said. "But as far as the mechanics and the amounts, we never talk about that stuff."
The Penguins re-signed forward Rickard Rakell to a six-year, $30 million contract ($5 million average annual value) Monday and defenseman Kris Letang to a six-year, $36.6 million contract ($6.1 million AAV) July 7.
Malkin is third in Penguins history in goals (444), assists (702) and points (1,146), and second in games played (981). He had 42 points (20 goals, 22 assists) in 41 games last season after missing the first half of the season recovering from offseason knee surgery.
He has missed 78 of the Penguins' 209 games since the start of the 2019-20 season.