Brawley City Manager Jerry Santillan said in January the City Council made a decision in public session that one of its goals for the year was to go back to a one-chief system.
He said the plan is to have a chief in place by year's end.
The city has started to advertise for the position regionally and he said the position will be advertised locally.
Santillan said while Graham and Kuhn will not be appointed outright, they both can apply for the position.
Still, he said, "Both of them are very qualified."
Neither Graham nor Kuhn was available for comment this morning. Kuhn has been absent from the department recently with a back injury.
Councilman John Benson said the council is interested in having one chief lead the department.
He said one key issue the council has wrestled with is should a search be done outside the department.
Benson said some council members are willing to appoint Graham while others want to review other applicants.
"I'm supporting looking out there because I think we need to always have the best person available," he said.
"I think Graham is a great man," he added.
He also said some council members think last year's controversy between the council and police officers could have been averted by having a chief.
That controversy revolved around a contract dispute and for a time officers cut back their services to the city. Officers also took the council and certain city officials to court, winning a legal battle over the contract.
The city eventually reached a contract settlement with officers.
Benson said the controversy was at least partly due to a breakdown in communication.
"We felt that at least a portion of our communication breakdown with the Police Department last year was because it was a divided leadership," he said.
Mayor Wayne Johnson agreed, stating, "I also feel what I call last year's mutiny was due to a lack of having a chief that was responsible to the City Council."
He said he supports advertising for the position, stating "there is a lot of talent out there."
He said he would not support Graham for the position.
"I have seen the way Graham operates and I don't agree with the way he operates," said Johnson, a retired Brawley police sergeant who worked under Graham.
Johnson said he does not think Kuhn is interested in the position.
The search for a chief is not expected to be discussed in Tuesday's City Council meeting, set for 7 p.m. in the council chambers, 225 A St.
Staff Writer Mario Renteria can be contacted at 337-3435. Staff Writer Darren Simon can be reached at 337-4082.