Houston City councillor Dave Martin told a district meeting: “I beg you not to send them a penny” and repeatedly branded the Congress-endorsed charity “the red loss”.
Speaking two weeks after the devastating hurricane which left at least 70 people dead, Mr Martin claimed that local government had done most of the difficult work in the aftermath of the disaster as well as providing the majority of resources.
“Yet every time I turn on the TV, I see (the Red Cross) taking in millions of dollars in donations,” he said, branding the charity “the most inept, unorganised organisation I’ve ever experienced”.
“Don’t waste your money. Give it to another cause,” added Mr Martin, an independent member of the council who has a background working in private sector management.
Other members of the council told local media after the meeting that they were grateful for the work of thousands of Red Cross volunteers across the region who have helped set up shelters for those forced to leave their homes after the hurricane caused severe flooding.
Other public officials have expressed concern about the organisation, including Harris County Judge Ed Emmett, who reportedly asked a local charity to set up a shelter for those displaced by the hurricane because he did not trust the Red Cross to do so.
“The Red Cross could not have done this. They wouldn't have had the wherewithal to do it,” Mr Emmett reportedly said. “Don't get me wrong, they're out there on the front lines, but I had already seen the difficulty and we needed to get this set up quickly.”
The charity has been criticised for failing to ensure supplies reached shelters quickly enough after one of Houston’s two Red Cross shelters could not accept evacuees at all because of high water levels and the other had just 200 beds for what turned out to be more than 2,000 people.
The shortages of beds continued for days while much of the city remained submerged.
A Red Cross spokeswoman defended the work of the organisation in Houston.
“We had all of our shelters on standby the night before Hurricane Harvey blew in, we had all our supplies ready and waiting to go,” MaryJane Mudd said in a statement.
“In some cases the floodwaters made it a little hard to get those supplies from where they were stationed into the shelters for a short while.
“We've had 1,500 people on the ground, we've served over 700,000 meals and snacks, we've sheltered 40,000 people. I know the plan was there. The process has worked very well.”
The American Red Cross, founded in 1881, has been hit by a series of controversies in recent years, including stinging criticism for the way in which it responded to the 2010 hurricane in Haiti and over Hurricane Katrina, which struck New Orleans in 2005.
Although not a government organisation, it is authorised to provide disaster relief since being granted a congressional charter in 1905.
It is part of the International Red Cross and Red Crescent Movement, which is believed to have around 100 million volunteers globally.