(CNN) It's an alarm immigrant rights groups have been sounding since the day President Trump took office.

Trump himself suggested well before his January 2017 inauguration that he planned to deport millions of people, telling "60 Minutes" days after his election that millions of criminals would be kicked out of the US once he was at the helm.

But while the number of deportations have increased since he took office, the mass deportations he warned of haven't materialized.

Trump's tweets Monday, threatening that "millions" of undocumented immigrants would be deported next week , revived fears that a major deportation operation was looming.

A day later, a senior immigration official with knowledge of the administration's plans painted a different picture. "There is no operation next week to pick up millions," the official told CNN. "No clue where he got that impression."

But Trump, who kicked off his 2020 re-election campaign with a rally Tuesday night in Orlando , stood by his tweets and maintained immigration officials were aware of the deportation initiative. "They know," he said. "They're going to start next week."

It's a familiar contradiction for Elaine Kamarck, a senior fellow at the Brookings Institution who co-authored a 2017 analysis of Trump's presidency titled "Hitting the wall: On immigration, campaign promises clash with policy realities."

"This whole area of immigration is one where there's just a vast gap," she said, "between the president's hysterical pronouncements and the reality on the ground and the capacity of the federal government."

Asking for a budget boost

According to Immigration and Customs Enforcement statistics, deportations increased about 13% between fiscal year 2017 and fiscal year 2018, when 256,085 people were deported. That's still significantly less than the number of people deported during fiscal year 2012, when President Barack Obama's administration deported more than 400,000 people.

Staffing limitations and budget constraints generally limit how many people the US can detain and deport -- and how quickly that process happens. Deporting millions of people would cost far more than Immigration and Customs Enforcement's current budget allows.

In an emergency budget request made last month , the White House is seeking $4.5 billion more from Congress "to address the immediate humanitarian and security crisis at the southern border of the United States," including $33.7 million more to fund ICE transportation and removal, and more than $340 million to fund thousands more beds in immigrant detention facilities.

But that request is still making its way through Congress -- and it's likely to face pushback, particularly from House Democrats who've repeatedly noted their concerns about the Trump administration's approach to immigration enforcement.

An 'atmosphere of fear'

ICE's acting director said earlier this month that the agency was exploring options to arrest and deport families who had gone through their legal proceedings and been ordered to depart the US. And the senior immigration official told CNN Tuesday that there is a tentative plan for a July operation that would target families who didn't show up at their immigration hearings and have removal orders.

Another official told CNN more than "1 million" undocumented immigrants "have been issued final deportation orders by federal judges yet remain at large in the country." That senior administration official called enforcing those judicial orders a "top priority" for ICE.

But it's unclear what will happen next.

Regardless, Trump's comments will likely instill more fear in communities that are already reeling, said Donald Kerwin, director of the Center for Migration Studies.

"The administration has created this atmosphere of fear in immigrant communities," he said. "Something like this is certainly designed to ratchet up the fear on the one hand, and on the other hand, appeal to people that want to see immigrants fearful."

Advocates argue that just because something hasn't happened yet doesn't mean it couldn't.

"ICE has been more than willing in the past to bust through its budget and find creative ways to pad their budget, overspend the money they're given and then go begging for more after they've spent it," said Ur Jaddou, a former chief counsel at US Citizenship and Immigration Services who now directs the watchdog group DHS Watch.

And authorities don't need to deport millions of people in order to send a message, Jaddou said.

"To sow fear and gain political points with his base, he only needs to say that, and then go after a lot more people than anyone has ever done, a lot more rooted families," Jaddou said. "All they need to do is go after a few thousand, and that will sow the fear that they are wanting to sew and gain the points that he wants to gain from his base."