Councilwoman Mary Fischer says the city mishandled negotiations from the beginning.
Mayor Tommy Roberts could not be reached for comment on Tuesday.
Johnson leases the restaurant space at Four Corners Regional Airport from the city and uses the city's liquor license to sell alcohol.
He requested contract changes including first right of refusal on a renewal and he also asked for changes to the lease's advertising policy during a meeting with city administration just over two weeks ago, said Bob Campbell, assistant city manager.
City staff is assembling the proposed changes in a report that will be presented to city council, he said. A date for that presentation has not yet been set.
Johnson seems to be satisfied with the process in the mean time.
"Everything's moving forward nicely and we have an open line of communication," he said in a telephone message.
Communication appears to be at the root of Fischer's concerns.
Reports on how the contract negotiations have developed should have been brought to city council for public review in a more timely fashion, she said.
"The whole process is being done backward," she said. "I seem to not be privy to the inner workings of the city. I think that they have handled the Zebra's situation with the misconception that staff is in charge of negotiations."
"When we're not told, how can we even have a position at all," she said. "It's just not done this way."
The situation could be unfair to Johnson as well, Fischer said.
"He's negotiating in good faith with people that do not have the final say," she said. "It shouldn't have taken but a couple of weeks to do this. It just doesn't seem that we get anything done."
Johnson could not be reached Tuesday for follow up questions about his negotiations with the city.
The Zebra's debate originated at a Sept. 25 city council meeting after Councilman Jason Sandel objected to a Zebra's billboard between Farmington and Kirtland depicting scantily clad women. He requested a review of the restaurant's lease agreement at the time.
Johnson agreed to take down the billboard after meeting with Campbell and airport manager Todd Gressick.
The negotiation process, however, has taken place largely behind closed doors, Fischer said.
In her opinion, it is time that these processes see the light of day.
"I have been here for a long time, and I've seen when (the city) worked," Fischer said. "It makes me sad to be part of a dysfunctional organization. It can be done. I've seen it done ... now I see this colossal mess on our hands."