President-elect Donald Trump stepped up his attacks Sunday against U.S. companies that move production offshore and fire workers.

In a posting on Twitter, he said there would be consequences like a 35% import tariff “for these companies wanting to sell their product, cars, A.C. units etc., back across the border. Please be forewarned prior to making a very expensive mistake!”

Instead of moving a factory outside the U.S., he encouraged business leaders to explore relocating factories between states and negotiate for tax breaks, encouraging a strategy that some state officials fear will escalate an arms race of incentives.

Mr. Trump helped fuel those concerns by engineering $7 million in tax breaks last week to save 800 jobs at a plant in Indiana owned by Carrier Corp., a unit of United Technologies Corp.

That deal could exacerbate a rush to grant tax incentives that have taken big chunks out of state budgets in some places. “Our prime competition used to be in Central America and Southeast Asia,” said South Carolina Republican state Sen. Wes Climer. “Now, some of our fiercest competition comes from neighboring states.”

Subsidies offered by states to companies have been on an upward trend for decades, apart from periodic economic downturns. An analysis by a professor at the University of Missouri-St. Louis about a decade ago found that states offered a total of $70 billion in subsidies annually.

Greg Albrecht, chief economist at the nonpartisan legislative fiscal office in Louisiana, said the Carrier deal could prompt some U.S. companies to threaten to move jobs to Mexico as a way to extract financial benefits. “I think we’re creating a moral hazard issue that’s bigger than we’ve got now,” he said.

In Louisiana, which hands out about $1 billion a year in annual tax exemptions to companies that meet certain targets, a $1 billion annual budget gap looms, he said.

“I truly have no idea what to expect,” said Tom Stanton, chief executive of telecom-gear maker Adtran Inc. “It’s unclear how what was said on the campaign trail is actually going to translate into legislation,” he said.

The company employs about 1,000 workers at its Huntsville, Ala., factory and mostly competes with overseas rivals. Asked about potential consequences for moving jobs offshore, Mr. Stanton said, “It’s probably better to incentivize than to punish.”

On Friday, Ford Motor Co. CEO Mark Fields said he would move ahead with plans to shift production of the Focus small car from a Michigan factory to a facility in Mexico, despite repeated criticism from Mr. Trump.

“We have made the decision to move the Focus out, and we’re making that investment now,” Mr. Fields said in an interview. “That’s to make room for new products—zero jobs affected, zero jobs impacted.”

On Friday, Mr. Trump announced that he would take economic counsel from a panel of business leaders that includes CEOs from Wal-Mart Stores Inc., General Motors Co. and private-equity firm Blackstone Group LP.

With the Carrier deal, Mr. Trump showed he was willing to make good on a populist campaign promise to keep jobs from leaving the country. With the CEO panel, Mr. Trump moved to confer with some of America’s top companies, many of which have vast international operations with thousands of employees overseas and are proponents of free and open markets.

“Free and fair global trade is important, and it definitely creates U.S. manufacturing jobs,” Boeing Co. CEO Dennis Muilenburg said after a speech in Chicago Friday. “We want to make sure that doesn’t get lost in the political rhetoric.”

For Todd Becker, CEO of Green Plains Inc., a Nebraska-based company that runs 17 ethanol plants, mainly in small U.S. towns, Mr. Trump’s actions show that “there’s a path in” for U.S. corporations.

“I like having a guy, who’s somebody who thinks like we do, in the White House,” Mr. Becker said.

In the Rust Belt, U.S. plants often tend to be outdated and need modernization. They are likely to use the threat of moving abroad to request “modernization money,” said Charles Bradford, an industry analyst. “I don’t think an American steelmaker has built a mill since the 1990s without major state help.”

Steelmakers including Nucor Corp. , AK Steel Holdings Corp. and Steel Dynamics Inc. have all received incentives from state governments. This year, Big River Steel started making metal at a new $1.1 billion plant built in northeastern Arkansas, thanks in large part to state aid.

Write to Scott Calvert at scott.calvert@wsj.com and John W. Miller at john.miller@wsj.com