WASHINGTON – President Obama moved to repair relations with a crucial voting bloc and opened another battle with Republican lawmakers by easing rules on illegal immigration.

The proposal likely could affect tens of thousands — perhaps more than 100,000 — illegal residents. It would end a requirement that undocumented immigrants with parents or spouses in the United States leave the country first if they wish to file paperwork that would forestall deportation on the grounds of family hardship.

Without the so-called hardship waiver, illegal immigrants are barred from re-entering the U.S. for up to 10 years. People seeking waivers often are separated from their families for months or, in some cases, years while their applications are processed.

Under the new rule, immigrants would be allowed to stay in the U.S. and apply for a waiver, which can be granted if deporting an immigrant would cause undue hardship to his or her U.S. family. Once waivers are granted, immigrants may apply for green cards. They would still have to leave the U.S. to make those applications, but because they’d have hardship waivers in hand, they’d have a strong likelihood of being able to gain readmission to the country.

Some 23,000 immigrants a year use the existing system. Many travel to Ciudad Juarez just over the border in Mexico — one of that country’s most dangerous cities — to file their applications and often are stuck there for extended periods. Administration officials expect many more people to apply for waivers once such trips become unnecessary.

The proposal left Republicans infuriated. Rep. Lamar Smith, R-Texas, chairman of the House Judiciary Committee, said the new rule, along with other moves by Obama, had “granted backdoor amnesty to potentially millions of illegal immigrants without a vote of Congress.”

Latino groups, many of which have criticized Obama for failing to move aggressively on immigration issues, were delighted. The move is a “sensible and compassionate proposal (that) helps bring much-needed sanity to an often senseless process,” said Janet Murguia, president of the National Council of La Raza, which describes itself as the nation’s largest Latino civil rights and advocacy group.