Some industry experts said the scope of the city’s order was unrealistic.

Anthony DeVito, vice president of engineering for Chemical Specifics Inc. in Maspeth, Queens, which maintains and cleans 500 to 1,000 cooling towers a year in the city, said his company was already straining to keep up with demand. He said that since the outbreak, he had received a number of calls from buildings around the city.

“Think Friday at 4 o’clock in the rain looking for a taxi,” Mr. DeVito said of the surge in demand he expected. “Everybody is going to have to clean these things at the same time. It’s pushing the limits of the available companies that do this kind of work to expect to complete this in two weeks.”

Mr. DeVito said it typically took two days to clean a cooling tower using a process that involved adding cleaning agents and sanitizers to the water circulating through the unit and then shutting it down to drain the water so that workers could scrub and pressure-wash the tower.

The Real Estate Board of New York, which represents building owners, said it was reviewing the order but supported the city’s efforts to address the outbreak. “Legionnaires’ disease is a serious issue, and building owners we’ve spoken to are responding appropriately,” said John Banks, president of the board.

Mr. de Blasio said the order would affect a limited number of buildings in the city because most did not have cooling towers. “They tend to be found in bigger, more modern buildings,” he said, “but any building that does have one of these cooling towers will be subject to this order.”

Experts who have long called for more stringent regulation of the towers said the outbreak and the city’s response offered a chance to address a serious public health risk.