BIRMINGHAM, Ala. — Veronica Roscoe, who earns $7.75 an hour working at a Burger King here, thought last August that she had won: This city had become the first in the South to approve a local minimum wage.

“It was, ‘Wow, victory,’ “ Ms. Roscoe, 55, said. “I was running around the job, acting a fool.”

But that was before the Alabama Legislature met, and before a showdown between state lawmakers and city leaders about who should have the authority to set wage policy in Birmingham. The dispute is a particularly ferocious version of a divide playing out nationally as cities increasingly move to raise their minimum wages and some states, particularly those controlled by Republicans, try to restrict their ability to set floors on pay.

The Alabama Senate is expected as soon as this week to consider a proposal, which the House approved overwhelmingly last week, that supporters believe would effectively end Birmingham’s ambitions for its own minimum wage of $10.10 an hour. Birmingham officials have reacted angrily and plan to consider a proposal on Tuesday that would put the city’s wage mandate into effect the next day, before Republicans could complete work on the bill making its way through the Legislature.

It is unclear what, exactly, will come after this week’s machinations, although partisan hostilities and legal arguments seem certain. At the least, the outcome is poised to reshape economic policy in this state and lead to refined strategies in the national debate about income inequality.