If he can’t, or if another company comes along with an offer too rich to refuse, Mr. Timken acknowledges what many in Canton would rather not say out loud. “If the number is big enough, the board is going to have to look at it and say, ‘Ugh, O.K., unless you can show me a plan that’s better than that, we’re going to take it,’ ” he said. “We’re a publicly traded company.”

As in all publicly traded companies, TimkenSteel’s board and top executives have a fiduciary duty to shareholders to maximize both profits and investor returns. For many companies in the region, that has meant cutting costs and moving production to places where labor is cheaper. Goodyear Tire & Rubber, for example, was founded in nearby Akron in 1898, but Goodyear hasn’t made regular tires in the onetime rubber capital of the world in more than three decades.

Timken briefly resorted to layoffs during the 2009 recession, but the company has been loyal to Canton. In the early 1980s, during a deep recession, Timken spent $450 million to build a state-of-the-art mill on the edge of town. More recently, the company spent $225 million to expand that steel mill and erect a giant caster, the only one of its kind in North America and one of only a handful anywhere in the world. Rising 17 stories above the surrounding cornfields with another eight below ground, the Jumbo Bloom Vertical Caster, as it’s formally known, can turn molten metal into fresh steel more quickly and at a lower cost. In October, it began making steel that will eventually end up as pipelines on deep-sea rigs, or as gun barrels or precision medical tools.

The Timken family supported these huge capital expenditures even though it meant lower profits in the short term and less capital to return to shareholders. The wager in the 1980s paid off in the long run, allowing Timken to innovate, dominate the market for high-margin, specialized steel, and stay ahead of rivals in South Korea, Japan and Germany. A chunk of the resulting profits has been poured back into Canton, a city of 70,000, in the form of good wages for unionized steelworkers or donations to local schools, the Canton Museum of Art, and the new downtown arts district. These efforts have largely spared Canton the fate of Youngstown, another Ohio steel town that is only now beginning to recover from 40 years of what economists politely term deindustrialization.

“I can’t imagine what Canton would be like without the Timkens,” said Kevin Dougherty, owner of the Industrial Tool Company, a Timken supplier. “I hope they maintain control, but it’s wait and see. You’d be hard-pressed to find somebody here that wasn’t provided for by that company.”

Adrian E. Allison, the superintendent of the Canton City School District, has a pretty good idea what it would mean if the Timken companies were sold. His mother worked for 30 years making vacuums at the Hoover factory in North Canton. That iconic company was sold by the Hoover family in the 1980s, beginning a long corporate odyssey through Maytag, Whirlpool and finally a Hong Kong company that closed the plant in 2007, sending the jobs to Mexico and China.

“It’s a legitimate concern within the community,” he said. “If the family left, the fear is it would end up like Hoover did.”