The council will then vote on his appointments.
The meeting will start at 6:30 p.m. in City Hall, 608 Heber Ave.
If the council decides not to remove Housing Authority directors, a second hearing may be set to consider replacing the entire board with the City Council.
The hearing is being conducted because of some calling for the removal of Housing Authority Executive Director Lupita Rios by way of the removal of board members who support her.
Rios cannot be removed by the City Council but can be removed by the housing board.
Six of the seven seats on the board are filled. Hildy Carrillo-Rivera stepped down earlier in the year.
Four of six board members favor keeping Rios, according to the president of the Housing Authority Resident Council, Maria Gasca.
Gasca said she will call for Alatorre's new housing board appointee to be someone who favors removing Rios and additionally will call for the removal of the board chairman, John Romo, for not removing Rios previously.
Rios has come under fire in recent months for a variety of charges ranging from discrimination against the low-income families she is hired to serve to intransigence in dealings with the U.S. Department of Housing and Urban Development.
Rios said Friday she will be at Tuesday's meeting to defend herself.
The City Council also on Tuesday will continue to prioritize the city's needs and wants to decide how to spend the $100,000 in economic development money the city received from the state.
Projects such as retention basins, security fences for the police and fire departments, financial support for the Small Business Development Center, improvements to the Community Center and new skate parks for youth will be discussed Tuesday.
Alatorre said prioritizing the projects will allow the city to pay for the most pressing needs without wasting money.
Mayor Pro Tem Victor Carrillo, who will become mayor in April, will see the completion of a personal project at Tuesday's meeting when the name of Topaz Court is officially changed to Bulldog Court
Carrillo made the promise to Calexico High School alumni and coaches who bought a number of houses on what became Topaz Court in housing developments on the northeast side of town.
At the last City Council meeting Carrillo was visibly irritated when he found out the name had not yet been changed.
Public works officials explained the process of changing a street name and vowed to have the paperwork ready for Tuesday's meeting.
Councilman Gilbert Grijalva has an issue with the Calexico Chronicle.
He will call for a vote of the council regarding whether the city should pull all advertising from the newspaper, which recently closed its Calexico office and moved operations to Holtville.
The reason for pulling the ads is a cartoon the Chronicle ran more than a month ago that Grijalva said stereotyped Mexicans.
Chronicle Publisher Steve Larson said he will not be blackmailed by the City Council.
"If they want to pull advertising, it is being very childish on their part and not serving the community on any basis," Larson said.