"We should know, I'm guessing in the next little bit with both players," Burke told "The Jeff Marek Show" Tuesday. "We've made determined efforts to meet with and to sign Kris Letang. Less determined with Malkin only because we have to know how much money we have to work with, and we have to put this jigsaw puzzle together.
"So very direct discussions with the agent for Evgeni Malkin but less intense. There's got to be some sequence here. One has to sign and then the other, or maybe neither of them. But it's coming to a head soon."
Malkin, a center, and Letang, a defenseman, can each become an unrestricted free agent July 13.
Malkin completed an eight-year, $76 million contact (annual average value $9.5 million) he signed June 13. 2013. Letang signed an eight-year contract, $58 million ($7.25 million AAV) July 3, 2013.
Burke reiterated that the Penguins want them back.
"We want to sign both players," Burke said. "Everyone should be emphatically clear on that. We would like to bring both players back, but they've got to come back at term and contract dollars that make sense. ... And if not, they're not coming back.
"That's what people should be very clear on. It's not that we're not interested in having them back. We want them back, but they've got to come back at numbers that make sense and don't leave us cap handcuffed. If that doesn't happen, they're not coming back. It's very simple for us. We want them back. They've got to come back at terms that make sense, or they won't be back."
Malkin, selected with the No. 2 pick in the 2004 NHL Draft, has played his entire 981-game NHL career for the Penguins, helping them win the Stanley Cup three times (2009, 2016, 2017). He won the Art Ross Trophy as the NHL regular season scoring leader twice (2009, 2012), the Hart Memorial Trophy voted as most valuable player in 2012 and the Conn Smythe Trophy as the MVP of the Stanley Cup Playoffs in 2009. The 35-year-old is third in Penguins history with 1,146 points, 444 goals and 702 assists, and second in games played (981) behind Sidney Crosby (1,108).
But Malkin has played more than 70 games in a season once since 2011-12. He missed 41 games this season after having knee surgery last June.
Letang was selected in the third round (No. 62) of the 2005 NHL Draft and has played his entire 941-game NHL career with Pittsburgh, winning three Stanley Cup titles. Also, 35, Letang is fifth in team history with 650 points and 506 assists and first among defensemen in goals (144), assists and points.
The Penguins were eliminated from the Eastern Conference First Round by the New York Rangers in seven games and have not won a playoff series since 2017-18.
After this season, general manager Ron Hextall said he wanted to see each player finish their NHL career with the Penguins.
"I think you look at them individually, they've both been here 16 years," Hextall said May 23. "[Malkin] has obviously been a great player, certainly one of the best players in the history of the game. We'd like to keep him as a Pittsburgh Penguin for the rest of his career.
"In a perfect world, 'Geno' retires a Penguin. I think 'Tanger' is the same. They've been here the same amount of time. These two, they're generational players. They don't come around very often."
Each has expressed playing several more seasons in the NHL.
"I understand it's a business," Malkin said May 17. "I love this city. I love these fans so much. If the team wants new blood, young guys, and say to me, 'You should move on,' I'm fine. … I hope I stay here forever, like retire here."
Letang said he could play another "four, five years. Easily."
"It's always been the main goal, to stay here and play in Pittsburgh," Letang said. "I certainly tried to build something good here. Just want to keep it going."
