“In a deeply polarized country, the minimum wage is one of a small handful of issues that gets broad bipartisan support,” wrote Dan Pfeiffer, who until recently was a senior adviser to President Obama, in an email in response to questions. “The Republican problem of opposing the minimum wage grows much worse when paired with their support of tax cuts for the wealthy and large corporations. The 30-second ad writes itself.”

To defuse the support, Republicans and pro-business groups have highlighted the potential job losses that could arise from minimum-wage increases. If “you just ask if people support a higher minimum wage, people support it,” said Michael Saltsman, research director of the business-backed Employment Policies Institute. “What’s interesting to me is when you have a follow-up question and walk them through the consequences, support really falls.”

A growing body of economic research suggests that moderate increases in the minimum wage would cause little or no job losses. An increase to $12 an hour spread out over the next few years, which would roughly restore the minimum wage to its purchasing power in the late 1960s, falls in this range, according to Jared Bernstein, who served as economic adviser to Vice President Joseph R. Biden Jr.

But moving substantially beyond that could lead to more significant job losses, particularly in low-wage regions of the country, said Mr. Bernstein.

The Senate bill would raise the minimum wage to $8 an hour in 2016, then in $1 annual increments until 2020. Thereafter, it would rise in line with the growth in median wages, which typically increase more rapidly than the inflation measure used in Social Security and other federal programs to adjust for changes in the cost of living.

Perhaps the most far-reaching element of the bill is its gradual elimination of a separate minimum wage for workers who receive tips, like waiters, which currently stands at $2.13. “This is historic, truly historic,” said Saru Jayaraman, co-director of Restaurant Opportunities Centers United and a leading activist on the issue.

According to Ms. Jayaraman, the practice of paying substantially lower wages to tipped workers dates back to the post-Civil War era, when dining establishments and railroads hired freed slaves to work for tips and nothing else.