According to Alex Marvez of Sporting News, O'Brien is prepared to broach the subject when he and Smith meet with McNair on Monday.
O'Brien is also reportedly expected to ask for say in the hiring of a new GM if McNair chooses to keep him over Smith with an eye toward having more input on personnel decisions.
On Friday, Aaron Wilson of the Houston Chronicle reported that sources said O'Brien and Smith have a "toxic" and "dysfunctional" relationship.
One source told Wilson they would be "kind of surprised" if the Texans decide to move forward with both O'Brien and Smith.
In four seasons together, O'Brien and Smith have led the Texans to two AFC South titles and one playoff win.
Houston went 9-7 in each of the previous three seasons and enter Sunday's game against the Indianapolis Colts with a 4-11 mark.
Injuries to defensive superstar J.J. Watt and rookie quarterback Deshaun Watson have contributed heavily to the Texans' struggles.
O'Brien could have some leverage to work with in his reported attempt to get Smith fired, as NFL.com's Ian Rapoport recently reported that the Arizona Cardinals could attempt to hire him as head coach if Bruce Arians does not return.
Overall, O'Brien is 31-32 in four seasons as an NFL head coach, but 2018 could represent his best chance to make a deep playoff run with the Texans provided both Watt and Watson are able to return to full health.