Declaring ‘‘now is the time’’ to fix the nation’s broken immigration system, President Barack Obama earlier on Tuesday outlined broad proposals for putting millions of illegal immigrants on a clear path to citizenship while cracking down on businesses that employ people illegally and tightening security at the borders. He hailed a bipartisan Senate group on a similar track but left unresolved key details that could derail the complex and emotional effort.

Potential Senate roadblocks center on how to structure the avenue to citizenship and on whether legislation would cover same-sex couples — and that’s all before a Senate measure could be debated, approved and sent to the Republican-controlled House where opposition is sure to be stronger.

Obama, who carried Nevada in the November election with heavy Hispanic support, praised the Senate push, saying Congress is showing ‘‘a genuine desire to get this done soon.’’ But mindful of previous immigrations efforts that have failed, he warned that the debate would be difficult and vowed to send his own legislation to Capitol Hill if lawmakers don’t act quickly.