McCown, 41, has played in the NFL for 19 seasons, primarily as a backup. He has taken the field for nine NFL teams, most recently for the Philadelphia Eagles in 2019 as a second-stringer to Carson Wentz.
The Texans signed McCown off the Eagles' practice squad last November, but the 19-year veteran did not take a snap in 2020.
Mike Garafolo of NFL Network tweeted about the candidacy of McCown, who does not have any NFL coaching experience.
"The Texans are exploring the idea of Josh McCown as head coach but would surround him with at least one former head coach on the staff," Garafolo wrote. "Again, not as crazy as it might seem to go from player to head coach. And if anybody can do it, McCown can."
Caldwell, 66, owns a 62-50 lifetime record in seven seasons overseeing the Colts (2009-2011) and Lions (2014-2017). Four of his seven teams went to the playoffs, including the 2009 Colts, who went 14-2 and won the AFC title.
The 14-2 record was the best mark achieved by a first-year NFL head coach since George Seifert went 15-1 with the 1989 San Francisco 49ers.
In addition, Caldwell was the first coach to guide the Lions to multiple winning seasons for the first time since Wayne Fontes in 1994 and 1995.
He most recently worked under head coach Brian Flores as the Miami Dolphins' assistant head coach and quarterbacks coach in 2019.