Adler has already, as of Friday, banned outdoor gatherings of more than 10 people, but he said he would like the governor to give local officials more leeway on additional restrictions that could be implemented as early as Wednesday.

"I don't know that the governor needs to do a statewide order at this point," Adler said. "But I want him to give us locally the ability to be able to do it, so that—so that I would have the ability to do that on Wednesday in my city."

Adler's office said Sunday night that the city will see new numbers from the University of Texas COVID-19 Modeling Consortium on Wednesday, along with an update from Austin-Travis County Interim Health Authority Dr. Mark Escott. The findings will be presented publicly Thursday.

Dr. Escott said last week that he would recommend a shutdown if the seven-day rolling average number of hospitalizations in the Austin area reaches 70. As of Sunday night, that average was 61.6.

Considering a stay-home order, if he can

Adler said Monday he is considering a stay-home order for the city due to the rising number of coronavirus cases and hospitalizations.

"It's something that we're considering," he said during a second appearance on CNN. "It's only to be used as a last resort."

As Adler said later in the interview, it's not clear that cities can institute stay-home orders without additional authority from Gov. Greg Abbott. "That's the conventional wisdom," he said.

Emma Freer contributed additional reporting.

This article has been updated with additional comments from Adler.