PRESIDENT OBAMA’S lopsided margin of victory among Hispanic voters in the presidential election alarmed Republicans and stirred hopes on Capitol Hill that an overhaul of the nation’s crippled immigration rules might just be possible. But as they prepare to dance their latest minuet on reforming a broken system, each party is starting with steps that the other finds unpalatable.

For the second time in three months, Republicans are pushing a bill that would create 55,000 additional green cards annually for foreign students who receive graduate degrees at U.S. universities in science, technology, engineering and math — so-called STEM graduates. On its own, that’s a fine idea, and a bipartisan one at that; as things stand, too many of these highly sought students are forced to leave the country upon graduation, to the detriment of U.S. competitiveness.

But the GOP measure, which was approved by the House of Representatives on Friday, comes at a cost: The 55,000 visas would be derived by eliminating a similar number of green cards disbursed by the so-called diversity lottery, a 20-year-old program that awards visas mainly to low-skilled Africans and Eastern Europeans. On top of having been a brilliant public diplomacy tool for the United States, the lottery has offered a lifeline for the sorts of immigrants who have energized this nation for decades by their pluck and drive.

Democrats are willing to expand STEM visas, but many are rightly reluctant to sacrifice the lottery. So the bill is likely dead on arrival in the Senate.

The White House and congressional Democrats worry that approving measures such as the STEM bill would make it more difficult to strike a deal later on a comprehensive immigration package. But in holding out for a perfect deal, they obstruct potentially good ones. Democrats agree that more STEM visas make sense; they’ve said so in their platform. They should seize the opportunity before them now — and seek a deal to keep some lottery visas.

What’s more, Republicans have sweetened the deal with a provision allowing the spouses and children of green-card holders, mainly from Mexico and the Caribbean, to enter the country more quickly while they await their own permanent-resident visas. That’s an improvement on the status quo, under which they must wait abroad to join their families, generally for two years or more.

Despite the hopes engendered by the election, the prospects remain cloudy at best for a comprehensive solution that satisfies everyone and resolves the central problem by offering a path to citizenship for 11 million illegal immigrants. While leaders in Washington try to crack that nut, it makes no sense to put other reasonable proposals on hold. Democrats and Republicans should be able to reach an accommodation on STEM visas. In poll after poll, Americans have said clearly they want the immigration mess cleaned up. So what is Congress waiting for?