O'Brien has one season remaining on the five-year contract he signed when taking over as head coach of the Texans in 2014.
Smith made the announcement in a statement less than an hour after the Texans finished their season with a 22-13 loss Sunday at Indianapolis.
"We have a bond here with the coaching staff and the players and maybe it's because of all the things that we kind of had to deal with this year", O'Brien said earlier this week.
Smith issued a statement through the team saying that he wants to focus on his wife, who recently was diagnosed with breast cancer.
"I am eternally grateful to the McNair family for their unwavering support during this trying time", Smith said. They would like to have the position filled this week, according to persons familiar with the situation.
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Another source speculated that Smith and O'Brien could keep working together but that it would be an "uneasy alliance" that's somewhat untenable.
It is unclear at this point whether O'Brien will have a greater say in personnel decisions in Smith's absence. He's had three 9-7 seasons and won two AFC South titles, but the Texans have collapsed to 4-11 this year after star rookie quarterback Deshaun Watson tore his ACL. Houston has traded away its first- and second-round picks in the 2018 NFL Draft and does not have a pick until the third round.
This comes after recent reports of a growing rift between coach Bill O'Brien and Smith.
It has been a season of struggle for the Texans in 2017.