Johnson, now an NFL studio analyst for Fox, tweeted early Friday: ""Looks like 2 of my guys getting NFL jobs..Chud Cleveland and my QB coach at U Trestman to Chicago."
Trestman was a quarterbacks coach in Johnson's regime at the University of Miami in the early 1980s.
It was news to Trestman, who wrote to the Montreal Gazette on Friday in response to the rumour:
"I have not heard from the Bears and have responded in that manner on multiple levels."
Johnson did adjust his comments several hours later: “I tweeted this morning ‘It LOOKS like 2 of my guys were getting NFL jobs not that Trestman got it.’”
The other reference in Johnson's tweet was to Rob Chudzinki being named as the head coach of the Cleveland Browns, a position for which Trestman also interviewed.
Trestman, who turns 57 next week, has guided the Alouettes to two Grey Cups since taking over in 2008, along with four seasons as the top team in the CFL East.
Prior to that, Trestman spent a quarter-century in American football at the pro and college level, working with NFL teams such as Oakland, San Francisco, Detroit, Miami and Arizona. A former quarterback in his college playing days, he's mentored such NFL pivots as Steve Young, Rich Gannon, Bernie Kosar and Jake Plummer.
Should Johnson be correct, Trestman will land in one of the more desirable, albeit pressure-filled, NFL coaching spots.
Lovie Smith was let go after nine seasons at the helm, but Chicago only missed the playoffs this season on a tiebreaker, finishing 10-6.
The Bears have not officially commented. According to other reports, they have not completed the interviewing of eligible candidates, including an African-American candidate as mandated by the NFL's so-called Rooney Rule.