Texas offered Bieniemy their running backs coaching position last week.
The position came open when the Longhorns' running backs coach, Michael Haywood, recently accepted the offensive coordinator position at Notre Dame.
Texas offered Bieniemy a sizeable increase in his salary at UCLA.
Late last week UCLA then matched the salary Texas had offered.
A factor in Bieniemy's decision was the potential care he could get for his 9-year-old son, Eric III, who has cerebral palsy. Texas reportedly has a good treatment center for the disorder, and offered potentially a better situation for the family.
UCLA, though, did what it could to provide the Bieniemy family an improved option in caring for their son. This, reportedly, was the deciding factor.
Sources indicate that Bieniemy, the fiery coach with a reputation as a great recruiter, always wanted to remain at UCLA, but that his first priority was the best situation for his son.
Bieniemy came to UCLA from Colorado, his alma mater. Having been an All-American running back for the Buffaloes in 1990 and having played in the NFL, Bieniemy had known Dorrell and came with him to UCLA last season. He is considered an excellent running backs coach and perhaps UCLA's best recruiter.