The motions were approved unanimously by the city's Planning and Land Use Management Committee. Both measures will now move to the full City Council for consideration.
The first motion would direct the city's Department of Building and Safety to withhold any new permits from Lineage Logistics to rebuild the facility before the fire department can fully investigate the fire.
The second motion would ask the city attorney to develop new fines that could be imposed on Lineage for violations and missed deadlines.
Lineage has already missed two deadlines to completely clear the site of rotting food following the fire.
As it stands now, the highest fine the city can issue is about $2,000 per violation. Last year, Lineage had revenue of more than $5 billion.
"We need to make sure we are putting all the punishment that we can on a company to hasten the process of cleaning up and being responsible," said Councilwoman Ysabel Jurado, who represents the 14th Council District where the warehouse is located.
Jurado said Lineage has indicated the cleanup will now be completed by Aug. 30.
Jurado hopes the two motions will next be considered by the full Los Angeles City Council later this week or next week.
As cleanup continues at the cold-storage warehouse following the June 17 fire, the stench of rotting food and pest problems have disrupted the surrounding neighborhood.
Company officials said they have already begun the next phase of the cleanup, which involves "proactive decontamination and cleaning in parallel with removing the remaining loose debris, including power washing, disinfecting, detailing and deodorizing."
"Our focus remains on fully remediating the site to eliminate odor impacts on the surrounding community," according to Lineage.
City News Service contributed to this report.