SAN JUAN, Puerto Rico—President Donald Trump commended rescue workers and passed out supplies to storm victims as he toured Puerto Rico on Tuesday, while also saying that the costs tied to Hurricane Maria have “thrown our budget a little out of whack,” drawing rebukes from Democrats who called the remarks callous.

During a visit to the island on Tuesday, Mr. Trump suggested that the death toll might have been much higher if not for exemplary rescue and recovery efforts. Officials told Mr. Trump in a briefing that 16 people had died in the storm. After he left, Puerto Rico’s governor announced that the tally had more than doubled, rising to 34, and would likely increase.

Mr. Trump drew a contrast with another deadly hurricane, Katrina, which devastated New Orleans in 2005 and killed more than 1,800 people. The president said that while “every death is a horror…you can be very proud of all your people—all of our people working together.”

The president’s visit came as the administration is expected to request more hurricane relief funding from Congress this week, according to Republican congressional aides, to replenish disaster-relief coffers as the nation recovers from being hit by three hurricanes in two months.

In their tour, the president and his wife, Melania, met with Puerto Rican officials before getting a firsthand look at the storm damage. While introducing his budget director, Mick Mulvaney, the president noted that the hurricane has strained the government’s coffers.

“I hate to tell you, Puerto Rico, but you’ve thrown our budget a little out of whack,” the president said. “Because we’ve spent a lot of money on Puerto Rico, and that’s fine. We’ve saved a lot of lives.”

President Donald Trump surveys the wreckage from Hurricane Maria on a walking tour with first lady Melania Trump in Guaynabo, Puerto Rico. Photo: jonathan ernst/Reuters

Mr. Trump’s remarks about the recovery costs angered Senate Democratic leader Chuck Schumer of New York.

“Mr. President, enough. Stop blaming Puerto Rico for the storm that devastated their shores, and roll up your sleeves and get the recovery on track,” Mr. Schumer said. “I don’t remember the president telling Texas that they threw our budget out of whack after Harvey. Or Florida after Irma.”

As of Tuesday morning, the unspent current disaster-relief fund stood at $9.25 billion. Of that, $2.03 billion is allocated to the areas affected by Hurricane Maria.

Lawmakers last month appropriated $15.25 billion in relief funds, two weeks after Hurricane Harvey hit Texas and while Hurricane Irma was making landfall in Florida. Hurricane Maria first made landfall in Puerto Rico on Sept. 20. An additional $6.7 billion became available on Oct. 1, stemming from Congress’s passage of a continuing resolution attached to the hurricane-relief measure.

Members of Congress on both sides of the aisle have signaled they would support legislation to help Puerto Rico and the U.S. Virgin Islands, which were hit by hurricanes in August and September, leaving residents with limited access to electricity, food and clean water.

“This is going to be a monthslong, yearslong effort,” said Rep. Carlos Curbelo (R., Fla.) “Can we do better? Yes. Should we do more? Definitely.”

In an interview that aired Tuesday night on Fox News, Mr. Trump also addressed Puerto Rico’s debt load. “We have to look at their whole debt structure,” he said. “You know, they owe a lot of money to your friends on Wall Street. And we’re going to have to wipe that out…You can say goodbye to that.”

In May, Puerto Rico was placed under court protection in what amounted to the largest-ever U.S. municipal bankruptcy. Puerto Rico and its agencies owed $73 billion to creditors.

Mr. Trump, wearing a dark windbreaker, spent part of Tuesday greeting residents and listening to accounts of the recovery effort.

President Donald Trump met San Juan Mayor Carmen Yulín Cruz, whom he had criticized on Twitter after she said the federal government’s response to the storm was inadequate. Photo: Evan Vucci/Associated Press

His first meeting brought him face-to-face with the San Juan mayor, Carmen Yulín Cruz, whom he criticized on Twitter after she had said the federal government’s response was jeopardizing lives. When the president and Ms. Cruz met Tuesday, they exchanged pleasantries, with Mr. Trump asking, “How are you?” and then thanking the mayor.

Later, asked by CNN about Mr. Trump’s comments on the budget impact, Ms. Cruz said it “doesn’t make you feel good” and showed his “lack of sensibility.” At the same time, she said she had productive conversations with Office of Management and Budget staff and the head of the Small Business Administration, Linda McMahon, who accompanied Mr. Trump on the trip.

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After his motorcade stopped at Calvary Church in Guaynabo, on the island’s northern coast, Mr. Trump passed out packages of rice and then pantomimed shooting baskets as he tossed paper-towel rolls into the crowd.

The governor, Ricardo Rosselló, said he planned to continue pushing for a federal aid package commensurate with what American citizens in Florida or Texas would receive. “My focus right now in terms of getting Puerto Rico pushing forward is to have an aid package that is consistent with the needs that Puerto Rico has right now,” he said. That could include such elements as lines of credit from the Treasury Department, he said.

GOP congressional aides say it is harder to make an estimate for the amount needed to pass a package for the islands because of the scope of the devastation and note the Federal Emergency Management Agency has access to the package that was passed earlier. But Democrats have been critical of what they say has been a slow response.

“There is no dollar estimate for how much money is needed because there is no plan in place,” said Sen. Elizabeth Warren (D., Mass.), who attended a briefing by FEMA on Monday.

In remarks ahead of Mr. Trump’s visit, Mr. Rosselló outlined progress the island is making in recovery efforts. Officials have bolstered the distribution system to deliver food and water to remote areas cut off by damaged roads and debris, he said. They dispatched additional trucks that can move provisions from regional staging sites to individual municipalities.

As the logistics effort gains more trucks and drivers, the movement of containers off the island’s ports increased to 855 on Monday, compared with a normal pre-storm daily flow of about 1,400, Mr. Rosselló said.

The telecommunications system continues to struggle. According the Federal Communications Commission, 88% of cell sites were out of service Monday. All of Puerto Rico’s 78 municipalities except for San Juan, Bayamón and Guaynabo had more than 75% of their cell sites down. And 27 had 100% of their cell sites out of service.

To address the breakdown, officials continued to set up mobile cellular stations across the island to provide temporary service for residents, Mr. Rosselló said. As a result, 40% of the island now fell under such coverage areas.

The island’s electric utility is working to reconnect the grid and get power flowing from its generators, many of which are in the southern part of the island, to customers. But on Tuesday, only about 7% of the electric utility’s customers had power. Officials continued to focus on restoring electricity to critical facilities such as hospitals and water plants.

Zoribel López, a 37-year-old customer service representative in San Juan, questioned how heartfelt Mr. Trump’s commitment was to Puerto Rico and whether he regarded the U.S. territory on par with the states.

“I think he truly believes we’re immigrants and we don’t deserve the same federal aid,” she said. She also worried about the long-term effects of the island’s economic paralysis and doubted the Trump administration’s policies would address them.

But Wilfredo García González, a 30-year-old computer-engineering student from the hard-hit town of Toa Baja, said he thought Mr. Trump had responded effectively to the storm. He stood outside the San Juan convention center—the nerve center for response efforts—wearing a “Make America Great Again” hat and carrying a sign that read in part: “make Puerto Rico great again!”

Write to Peter Nicholas at peter.nicholas@wsj.com, Arian Campo-Flores at arian.campo-flores@wsj.com and Natalie Andrews at Natalie.Andrews@wsj.com