Klemm was suspended for alleged NCAA violations.
The review continues but Klemm is being allowed to resume his duties as the offensive line coach.
Klemm has been an assistant at UCLA for the past three seasons and is also one of the school's top three recruiters and has rebuilt the Bruins' offensive line.
He missed spring ball and was not allowed to recruit after being suspended in March.
A person familiar with the situation said Klemm's only likely punishment will be the time he has already missed, including spring practice.
UCLA said they will refrain from comment to "protect the integrity of the ongoing process."