Forsberg, who has played his entire 10-season NHL career with the Predators, had career bests in goals (42), assists (42) and points (84) in 69 games this season, the last of a six-year, $36 million contract ($6 million average annual value) he signed June 27, 2016.
He is Nashville's all-time leader with 220 goals and is fourth with 469 points.
Forsberg said his goal is to re-sign with Nashville when he spoke to the media May 10, after the Predators were eliminated from the Stanley Cup Playoffs, swept by the Colorado Avalanche in the Western Conference First Round.
"We're so close right now to the deadline that we're all going to know pretty soon," Poile said. "We just keep pecking away. We're just putting our best foot forward."
Poile said he's hopeful that the Predators' trade for defenseman Ryan McDonagh on Sunday further entices Forsberg to want to come back to Nashville.
The Predators acquired McDonagh from the Tampa Bay Lightning for forward Grant Mismash and defenseman Philippe Myers. The 33-year-old won the Stanley Cup with the Lightning in 2020 and 2021 and helped them get back to the Stanley Cup Final this season, when they lost to the Colorado Avalanche in six games.
"A lot of our players reached out to me and said how happy they were that we were able to acquire him, and that's going to be a big boost for our team," Poile said. "Filip, I would hope he would be encouraged that that's a move in the right direction for what we're trying to do, and that's win hockey games."
The Predators will be active in the free agent market July 13 if Forsberg doesn't re-sign with them, Poile said.
"We're not going to be shut out," he said. "We're going to have Filip or we're going to do our best to be busy on the 13th. If it's not done by the 12th then he's free to go wherever he wants, so on the 13th that's our opportunity, but the first priority, still our priority, our No. 1 is to try to get him signed."