Lawmakers are trying to fast-track a bipartisan bill to boost carbon capture projects, but it isn’t clear if the legislation will make it to the president’s desk this year.

Timing is crucial for the bill, S. 2602, which builds on a bipartisan extension and revision of tax credits for carbon capture and storage projects included in March’s congressional spending deal.

The carbon-capture push has garnered significant bipartisan support in the Senate, but if the Democratic Party takes either—or...