(CNN) South Florida's Miami-Dade County has long been known as a place that welcomes immigrants, but its Republican mayor has ordered jails there to "fully cooperate" with President Donald Trump's crackdown on so-called sanctuary cities.

Mayor Carlos Gimenez sent a memo Thursday instructing the county's interim corrections director to comply with all immigration detainer requests received form the Department of Homeland Security.

Sanctuary cities have policies in place designed to limit cooperation with or involvement in federal immigration enforcement actions.

The mayor's directive comes as several Democratic mayors in the nation's largest cities have banded together to fight the Trump executive order that White House press secretary Sean Spicer said will "strip federal grant money from the sanctuary states and cities that harbor illegal immigrants." The President reiterated his desire to target such cities Thursday when he spoke to a congressional Republican retreat.

"Miami-Dade County complies with federal law and intends to fully cooperate with the federal government," Gimenez wrote in the memo.

Since 2013, Miami-Dade, the state's largest county in terms of population, has had a policy of not holding detainees who may be in the country illegally unless the costs of their detention are reimbursed by the federal government.

'Never considered ourselves a sanctuary city'

Still, the county -- which has the second highest number of immigrants in the country, with more than 1.3 million, according to the Migration Policy Institute -- has long objected to being classified as a sanctuary city.

"We never considered ourselves to be a sanctuary city," Gimenez told CNN Friday. "Miami-Dade County has never withheld information from the federal government."

Gimenez said his order eliminates a requirement that the federal government agree to reimburse the county for detention costs.

"It's really not worth the risk of losing millions of dollars to the residents of Miami-Dade County in discretionary money from the feds," said Gimenez, who is himself a Cuban immigrant, born in Havana.

"So, in a case where it doesn't really amount to much here ... I made the decision to no longer require the federal government to give us a document indicating that they will pay the cost for that inmate."

Friday, dozens of demonstrators gathered outside the Miami-Dade County government center to protest the mayor's memo.

The American Civil Liberties Union of Florida Thursday expressed disappointment with the mayor's order, saying such policies "serve only to drive a wedge of distrust between law enforcement and our immigrant community."

"At the very least, a warrant from a court, not merely a request from a federal official, is required to detain somebody in jail," the ACLU said in a statement.

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Some mayors plan to challenge executive order

Mayors in cities including Los Angeles, Boston and New York, as well as legal scholars, have vowed to challenge the presidential order, saying Supreme Court cases makes it difficult for Washington to punitively withdraw money from state and local governments.

"We feel very strongly that the legal case is clear," Los Angeles Mayor Eric Garcetti told reporters after the executive order was announced.

He alluded to NFIB vs. Sebelius, a 2012 case that challenged aspects of former President Barack Obama's health care law. In one part of that decision, seven justices including Chief Justice John Roberts, agreed the Medicaid expansion provision under the ACA violated the Constitution by threatening to take away states' Medicaid money if they refused to comply with the expansion.

"Back then it was a different politics, maybe it's ideologically reversed, but it was loud and clear," Garcetti said. "You can take away funding from a specific program if you don't adhere to the requirements of that program, but we don't have funding that is for the cooperation of our immigration federal officials and our local officials."

Karen Tumlin, legal director for the National Immigrant Law Center, said there is a long line of cases that deal with the spending clause of the US Constitution. They have underscored, she said, that the federal government can put some requirements on states when they parcel out funding, "but it can't be unduly coercive."

What does 'sanctuary city' mean?

Enforcement of Trump's executive order is complicated by the fact that the term "sanctuary city" has no universal meaning.

Police chiefs around the country have widely varying policies in the degree to which they cooperate with Immigration and Customs Enforcement agents and the lengths to which they will go to protect undocumented immigrants.

Law enforcement leaders in some cities have argued it is invaluable to have a strong relationship with immigrant communities -- and limit fear of deportation -- when they are trying to solve crimes. Undocumented immigrants, they note, can serve as helpful informants.

Democratic mayors are being forced to balance the need to maintain a good relationship with the Trump administration while at the same time answer to constituents seeking a path to citizenship for as many as 11 million immigrants living in the United States.

Boston Mayor Marty Walsh was firm in his resistance this week -- promising that if necessary, he would "use City Hall itself to shelter and protect anyone who is targeted unjustly."