By Mark Peters And Rachel Feintzeig 

As corporations, public officials and even a rock band ratcheted up pressure on Indiana over its days-old religious-objections law, state lawmakers on Monday said they would try to find a way to clarify that the measure wouldn't allow discrimination against gays and lesbians.

Engine maker Cummins Inc., based in Columbus, Ind., along with Dow AgroSciences LLC and Roche Diagnostics USA, both based in Indianapolis, said they wrote Republican Gov. Mike Pence and state lawmakers encouraging them to make changes to the law. Rock band Wilco said it was canceling a May concert in Indiana. Connecticut Gov. Dannel Malloy put a stop to state-funded travel to Indiana, including those affiliated with the University of Connecticut.

The new law sets a legal framework for people or companies to challenge government rules that hamper their practice of religion. Opponents, and even some supporters, of the measure say it could allow for businesses, including wedding-service providers, to seek court protection not to provide services to gays and lesbians.

State lawmakers in Indianapolis, which is preparing for this weekend's NCAA Final Four basketball tournament, are discussing crafting a bill to clarify the law that could come later this week.

Republican House Speaker Brian Bosma said the bill wouldn't allow discrimination, and added the governor's refusal to clarify that helped fuel the controversy. Mr. Bosma said lawmakers plan to make that "crystal clear" in the new bill they are drafting.

The law comes a year after the state legislature failed to get a constitutional ban on same-sex marriage on the ballot, which angered social conservatives. Some have described the new law as an effort to make up for that defeat. A federal circuit court ruling allowed for gay marriage to begin in Indiana last fall, while a key Supreme Court ruling on the issue is expected this summer.

While the sudden national attention on the state caught some off-guard, others were less surprised after working for months to defeat the measure. Marya Rose, chief administrative officer at Cummins, remembers a protest more than a decade ago at a shareholder meeting after the company provided benefits to same-sex employees. But such opposition has long been fading.

"People were just not in tune with how much people have changed about how they think about this," she said.

Polling done last year by the Pew Research Center found support for same-sex marriage has been steadily growing, with 54% of those surveyed favoring such unions compared with 31% a decade ago.

As for wedding-service businesses, the survey found 49% of those surveyed said they should be required to serve same-sex couples, while 47% said such businesses should be allowed to refuse services on religious grounds.

For years, large-company chiefs worked behind the scenes on controversial issues, quietly lobbying those in power while maintaining public silence, says Sydney Finkelstein, a professor of strategy and leadership at the Tuck School of Business at Dartmouth.

That has been changing, especially among tech companies. Some of the strongest opposition has come from that community, including from Apple Inc. Chief Executive Tim Cook and Angie's List Inc. CEO William Oesterle, who had managed the 2004 campaign of popular former Gov. Mitch Daniels, a Republican.

Still, some find the timing of the outpouring questionable.

"I knew there would be some pushback, but I had no idea" how large it would get, said Curtis Smith, president of the Indiana Family Institute, a conservative group that lobbied in support of the law. He said the bill had garnered very little attention until it was signed.

The original federal Religious Freedom Restoration Act, passed in 1993 with broad, bipartisan support, had little to do with gays and lesbians. Rather, it was largely aimed at shoring up protections for less widely practiced religions whose unusual rituals, like smoking peyote during ceremonies, might be rendered illegal by laws of "general applicability, " like bans on psychoactive drugs. The law passed in 1993 and was signed into law by President Bill Clinton.

While bills such as the one in Indiana have been proposed in more than a dozen states this year, few appear on track to become law. In recent days the Montana legislature rejected a statewide vote on such a law, while in Georgia, such legislation passed the Senate this month but stalled in a House committee last week.

In Arkansas, lawmakers could vote on a similar bill as early as Tuesday. Gov. Asa Hutchinson, a Republican, has promised to sign the bill into law, but added he is "pleased that the legislature is continuing to look at ways to assure balance and fairness in the legislation."

The challenge for Indiana legislative leaders is building support for immediate changes to a law that is less than a week old. The legislation passed overwhelmingly in the Republican legislature.

Mr. Bosma and Senate President Pro Tem David Long said they are working with their caucuses and Mr. Pence on the proposed language of what they are describing as a clarifying measure. They said they are looking to move quickly on the legislation.

Support for the bill from Democratic leaders appeared unlikely. They are pushing instead for a repeal of the law rather than a clarification. At a morning news conference, they said Republicans aren't moving aggressively enough in response to the opposition.

"They were wrong, and it's time they admitted they made a horrible mistake," said state Rep. Scott Pelath, Democratic minority leader.

Write to Mark Peters at mark.peters@wsj.com and Rachel Feintzeig at rachel.feintzeig@wsj.com

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