The Texans also announced they have "continued discussions" about the vacancy with Caldwell.
Caldwell, 66, most recently was hired as an assistant head coach with the Miami Dolphins in 2019 but had to leave the team ahead of the season to address health issues and was not retained for the following season. He went 36-30, including playoffs, as head coach of the Lions (2014-17) and 28-24 as head coach of the Colts (2009-11).
McCown, 41, retired in 2019 after 17 years in the NFL and took a job as a high school quarterbacks coach in Charlotte, North Carolina, where his sons played. He continued to coach after coming out of retirement and signing with the Eagles later in 2019.
When McCown signed with the Texans, he was asked whether he hoped to coach in the future.
"I believe so," McCown said in November. "If you'd asked me that 10 years ago, I would've said no way, that's not what I want to do at this level. My dream coming out of college had I not been able to play in the NFL was to coach high school football here in Texas. I really thought that was going to be my path. But the longer I've been in this game and been around these guys, I see real value in being able to be a part of an organization and compete at the highest level. ...
"Yeah, one of these days, whenever that happens, whenever I finally take the cleats off, I definitely see that in the future."
The team announced earlier in the week that it had interviewed Bills defensive coordinator Leslie Frazier, Chiefs offensive coordinator Eric Bieniemy, Colts defensive coordinator Matt Eberflus and Ravens assistant head coach David Culley.