Mitt Romney, hunting for an electoral edge in swing states, is intensifying his push for Hispanic voters, ratcheting up his Spanish-language advertising, deploying a Spanish-speaking son to court Latino leaders and putting himself in front of a growing number of Hispanic audiences.

On Wednesday, Mr. Romney brought his conspicuous outreach to Miami, where he participated in a candidate forum hosted by Univision, the dominant Spanish-language television network in the country, and attended a late night “Juntos con Romney” (“Together With Romney”) rally.

Mr. Romney’s brutal primary campaign at times put him at odds with quarters of the Hispanic community, a fact that the hosts of the Univision forum did not shy from. They posed pointed questions about illegal immigrants and whether the Spanish language has a place in American life. (“Spanish,” Mr. Romney said, quoting a friend, “is the language of our heritage. English is the language of opportunity.”)

Despite repeated inquiries, Mr. Romney avoided saying whether he would continue a program to suspend deportations of young illegal immigrants announced in June by Mr. Obama. Instead he accused the president of using immigration as a “political football,” and he returned to a promise to “put in place a permanent solution” to illegal immigration.

Pressed for details, Mr. Romney said again that he would support giving legal permanent residence to illegal immigrants who serve in the military. But he also suggested he would support another big piece of a bill in Congress known as the Dream Act. “Kids that get higher education could get permanent residence,” Mr. Romney said, in what appeared to be another step away from his position during the nominating contests, when he said he opposed the Dream Act.

Mr. Romney rejected mass deportation of illegal immigrants, but he sidestepped a question about whether he still supported encouraging “self-deportation”—encouraging such immigrants to leave the country by strictly enforcing immigration rules, a position he has advocated before.

“We are not going to round up people around the country and deport them,” he said. “Our system is not to deport people.”

Alberto Martinez, an adviser, said the Romney campaign was organizing “the most aggressive Hispanic outreach of any Republican presidential campaign.”

But during a rally on Wednesday night in Miami, surrounded by Spanish signs and introduced by his [Romney’s] son Craig in Spanish, he argued that the Republican Party had earned the affections of Hispanic voters.

“This party,” he said, “is the natural home for Hispanic-Americans.”