"With the situation the program is in, for a coach to be interested in the job it's not going to be about money," Barnett said Sunday during a phone conversation.
"It has to be somebody that is willing to go through the rebuilding process and accept the fact that he will start out with losing seasons."
The new coach must scramble to put together a recruiting class prior to the Feb. 6 national singing day. But Barnett said what will be even more critical for the next staff is to convince players already in the program to stay.
Some attrition is inevitable with any coaching change. But Colorado, despite its 1-11 record, does have some young talent and can't afford to lose it.
"If you miss on some recruits now, what you'll be missing is next year's freshmen," Barnett said. "You can replace that with the following year's recruits.
"But if you lose guys that have been in the program, guys you've invested years in, they're impossible to replace. That would set the program back even more."
Colorado received some good news Sunday when its lone quarterback commitment, Sefo Liufau of Tacoma, Wash., reconfirmed his allegiance to the Buffs during an interview with Rivals.com.
As a senior, Liufau threw for 2,518 yards and 24 touchdowns, against just five interceptions. He completed 69 percent of his passes, according to Rivals.com.
Liufau said he has been hearing from other colleges since CU fired Embree but is remaining patient while the Buffs program looks for a new coach.
"It is kind of the business end of things," Liufau told Rivals.com when asked about CU's coaching change. "I'm just trying to see what happens because I still love Colorado and I would still love to go there."
Meanwhile, Fresno State hosted two CU commitments for visits during the weekend. Both are wide receivers: Elijah Dunston (6-foot-1, 180 pounds) from West Hills (Calif.) Chaminade High School and Devin Ross (5-9, 170) from Mission Hills (Calif.) Bishop Alemany.