Taylor injured his left hamstring in the first half in the Texans' 31-21 Week 2 loss to the Cleveland Browns and was replaced by Mills in the second half. The Texans used their top pick, a third-round selection, on Mills in April.
Taylor was placed on injured reserve Tuesday, meaning he will miss at least the next three games.
In his NFL debut on Sunday, Mills completed 8 of 18 passes for 102 yards with a touchdown pass and an interception in the loss, which dropped the Texans to 1-1.
When Culley made the announcement, he said he thought that Mills would be more comfortable on Thursday because he will be able to take the majority of the first-team reps Tuesday.
"He's been ready," Culley said of Mills on Tuesday. "Now that he's gotten those reps, I think there will be a comfort level in starting the game. I think in the second half when he had to go out and play, I think not having had those reps in practice and all of a sudden now, 'I'm up,' probably made a difference. And I think this week ... when we start the game and he's out there, I think there will be a much better comfort level."
On Monday, Culley said quarterback Deshaun Watson would not be active Thursday night, just as he was not for Houston's first two games this season. Watson faces 22 active lawsuits with allegations of sexual assault and inappropriate behavior, but he has not been restricted from taking part in team activities by the NFL.
The quarterback also requested a trade in January because he was not happy with the direction of the franchise and the process used to hire general manager Nick Caserio.
Culley said the Texans will likely call up quarterback Jeff Driskel from the practice squad to back up Mills on Thursday.