Asked in an interview if he would be willing to change the Jeep Cherokee’s name if pushed to do so, Mr. Tavares said, “We are ready to go to any point, up to the point where we decide with the appropriate people and with no intermediaries.”

“At this stage, I don’t know if there is a real problem. But if there is one, well, of course we will solve it,” Mr. Tavares said, adding that he wasn’t personally involved in the talks.

Debate over the Cherokee name is among the issues facing Mr. Tavares, who took control of Stellantis when it was formed earlier this year from the merger of Fiat Chrysler Automobiles NV and Peugeot-maker PSA. In the interview Wednesday, Mr. Tavares also discussed whether to cut down on the company’s 14 brands, making Fiat plants more competitive and his plan to stick with China.

The Cherokee Nation is the largest Native American tribe in the U.S., with some 370,000 members, and Jeep has sold millions of vehicles named after it. The auto brand extended its use of the Cherokee name to a compact SUV, a smaller version of the Grand Cherokee, in 2013.

The leader of the Cherokee Nation recently said he would like to see Jeep stop using his tribe’s name on its SUVs.

Chuck Hoskin Jr.,

principal chief of the Cherokee Nation, said that he believed Jeep had good intentions but that “it does not honor us by having our name plastered on the side of a car,” according to a statement first released to Car and Driver last week.

“The Cherokee Nation has an open dialogue with Stellantis leadership, and look forward to ongoing discussions,” a spokesman for the tribe said Wednesday. “We appreciate Stellantis’ reaching out and thoughtful approach on this.”

Mr. Tavares’s remarks come in the wake of a broad reckoning over racial and social injustice in the U.S. that was sparked by the police killing of

George Floyd,

an unarmed Black man, in Minneapolis over Memorial Day weekend last year. In December, the Cleveland Indians decided to drop the baseball team’s longtime nickname after fans and Native American groups criticized it as racist. The Washington Football Team of the NFL has dropped a name that had been seen as a racial slur.

The Jeep Cherokee and Grand Cherokee SUVs are among the brand’s bestsellers in the U.S., accounting for 43% of Jeep’s sales in its largest market, according to company figures. Stellantis is rolling out a long-awaited redesign of the Grand Cherokee later this year.

Mr. Tavares said the auto industry’s practice of naming cars after Native American tribes was a sign of respect.

“I don’t see anything that would be negative here. I think it’s just a matter of expressing our creative passion, our artistic capabilities,” Mr. Tavares said.

The Jeep brand sits alongside profit-drivers like Ram in the U.S. and Peugeot in Europe. But the company’s sprawling portfolio of 14 brands also includes some that will need to prove their worth, Mr. Tavares said.

Mr. Tavares said he has asked each of his brand chiefs to work on a 10-year plan to develop more long-term visibility on product planning.

“I’m saying, ‘Look guys, I’m going to give you a chance. You need to convince me—you, the brand CEO—that you have a vision,’” Mr. Tavares said.

After several turnaround efforts, Fiat Chrysler’s Alfa Romeo and Maserati brands have failed to mount meaningful comebacks in recent years. The Fiat brand struggles with aging models and weak sales, which has caused an overcapacity problem in the company’s Italian factories.

Even the storied Chrysler brand has waned in recent years, now selling only three models compared with the six it carried a decade ago. The brand’s U.S. sales have also slid to one-third their volume in 2015, according to company figures.

On the PSA side, the DS brand—which focuses on high-end sedans and SUVs—grew market share last year but continues to lag far behind some of its German competitors.

“After we give them a chance to fail, we need to be also fair,” Mr. Tavares said. “If the rest of the company is doing the right things and there is one part of the company that is pulling everybody down, we’ll have to take that into consideration.”

The Portuguese executive built his reputation in the automotive industry as a turnaround expert. Peugeot was bleeding money when it hired Mr. Tavares in 2013. Since then the French car maker has gone from losing 5 billion euros, equivalent to about $6 billion, in 2012 to becoming one of the most profitable mass-market car makers in the industry. Last year it reported a net profit of €2.17 billion, or roughly $2.62 billion, with an adjusted operating margin of 7.1% in its core automotive business.

This time, Mr. Tavares has a longer to-do list, including integrating the two companies’ European businesses and stemming losses in China.

In Europe, Mr. Tavares has been visiting Fiat Chrysler factories—including an Alfa Romeo facility 80 miles south of Rome—and encouraging them to benchmark their performance against PSA plants. Additionally, employees from Fiat Chrysler’s Fiat factory in Mirafiori, Italy, visited PSA’s Citroën’s plant in Madrid, and Mr. Tavares said they were surprised by the nonlabor cost savings they observed.

The auto executive said the new company could reach its cost-saving goals in Europe without closing factories.

Asked what lessons he had learned from the chip shortage that has idled car plants across the world, Mr. Tavares said large suppliers didn’t relay signals they were receiving about the looming crisis. “We were not protected,” he said. “That’s a clear lesson learned.”

Mr. Tavares said the industrywide shift toward electrification would continue to rely on government subsidies and other financial incentives for buyers until auto makers figure out how to lower production costs over the next few years.

“If we propose electric vehicles which are extremely efficient but nobody can buy because they are costly, what’s the point from an environmental perspective?” he said.

In China, the combined sales of Peugeot and Fiat Chrysler accounted for less than 1% of a market that sold 20 million vehicles last year, according to industry data. Fiat Chrysler has long struggled to turn a profit in the world’s largest automotive market, while the French car maker sold only 45,965 vehicles in China last year, continuing a rapid multiyear decline.

Mr. Tavares said Stellantis isn’t considering exiting China, removing an option that he said was still on the table when the company started trading in New York at the start of this year.

“We cannot be away from the biggest market in the world,” he said.

Write to Nick Kostov at Nick.Kostov@wsj.com and Nora Naughton at Nora.Naughton@wsj.com