Kory Clemens, 30, is being charged with DWI 1st offense, a misdemeanor.
Around 2:15 a.m., officers were on a separate traffic stop when they saw a black Ford F-150 speeding southbound on Hillcroft.
Police caught up with the truck at the corner of Bellaire and Hillcroft. The driver, identified as Kory Clemens, then turned left while the turn lane light was still red.
Officials stopped Kory, who they say showed signs of intoxication.
Kory was booked into the Harris County Joint Processing Center. Authorities say Kory had a passenger with him.
It's not known whether that person will be charged.
Kory is the second eldest of Clemens' four sons. Like all of his brothers, Kory has connections to the University of Texas at Austin, attending the school before launching a career in culinary arts.
He graduated from culinary school at the age of 23 and was trained in French cuisine.
Kory's father said at the time that he also interned at several restaurants, with stints at Ragin' Cajun and the Taste of Texas and pursued restaurant management.
In 2013, that path led Kory to the opening of a new Houston restaurant called Katch 22, where he was the chef.
The restaurant opened on May 15, 2013, the 29th anniversary of Roger Clemens' major league debut. The restaurant shuttered a little over a year later.
In January of this year, another son of Roger Clemens filed a lawsuit against a Houston bar, alleging he and his godbrother were unjustly assaulted by bouncers and staff on New Year's Eve.
Kacy Clemens was with Conner Capel when they claim workers at the Concrete Cowboy singled them out by a bouncer who told them to move.
The men complied, but the bouncers then tried to throw them out of the bar, the lawsuit alleges.
Kacy and Capel are both baseball players in the minor league.
READ MORE: Roger Clemens' son and godson suing bar over alleged assault by bouncer on New Year's Eve