COLUMBUS, Ohio — Get live updates about the Ohio primary.

Two weeks ago, Gov. John Kasich was facing pressure to quit the Republican race, to clear the establishment field for Marco Rubio as the Florida senator sought to rally Republican voters in mounting a challenge to stop Donald J. Trump.

Now, with Mr. Rubio’s campaign unraveling, Mr. Kasich has a chance to seize his place as the main establishment alternative to Mr. Trump, one who can soak up Republican votes among people determined to find a more electable candidate to challenge Hillary Clinton should she become the Democratic nominee.

But it could be a tough fight. Mr. Kasich has failed to win a single nominating contest, and missed a big opportunity to gain attention this week when he placed third in Michigan, behind Senator Ted Cruz of Texas, coming up short in a state the governor had lavished with attention.

On Tuesday, he will have a final chance to establish himself as a viable contender in a shrinking field, when his home state, Ohio, has its winner-take-all primary, one of the biggest prizes on the electoral map. His campaign is betting that his strong organization in the state will propel him to victory, making him a force as the contest turns to more moderate Northern states, including a slew of Northeastern states that hold their contests in April.