On Tuesday, Ken Rosenthal of The Athletic reported the Jays "seem destined to move on" from Gibbons, who started his second stint with the club in November 2012.
Toronto holds a 51-60 record, which leaves the club 27 games behind the Boston Red Sox in the AL East and 15 games behind the AL's final wild-card spot, currently held by the Oakland Athletics.
Gibbons first joined the organization's staff in 2002 as a bullpen catcher. He was promoted to manager ahead of the 2004 season and held the position through June 2008.
The 56-year-old Montana native went on to join the Kansas City Royals staff in 2009 and spent time managing the San Antonio Missions at the Double-A level before returning to the Blue Jays.
Toronto made back-to-back appearances in the American League Championship Series in 2015 and 2016 but couldn't capture the team's first World Series berth since 1993. The Blue Jays missed the playoffs last season.
In June, Gibbons reached a milestone by managing his 1,500th game.
"It means I've been around a little bit. I'm proud of that," he told reporters. "I got some breaks too. That's what you gotta have in this game. A lot of really good baseball guys don't get them, and I got them."
Although the Jays are only a couple of years removed from championship contention despite being stuck in the same division as the Red Sox and New York Yankees, it appears they are preparing to move in a new direction at manager for 2019, barring a miraculous postseason surge over the next two months.