WASHINGTON—An 11th-hour push by Democrats to fund their social-spending and climate bill faltered, with a proposed tax on billionaires’ unrealized capital gains and revised bank-reporting requirements both running into opposition from party lawmakers.

Democrats have scrambled for days to find hundreds of billions of dollars worth of new revenue acceptable to Sen. Kyrsten Sinema (D., Ariz.), who is opposed to the increases in the top marginal rates on companies, capital gains and personal income that the party planned to rely on to finance the bill. With Republicans opposed to the package, Democrats need unanimous support to pass the legislation in the 50-50 Senate and can afford to lose only a few votes in the House.