Pence told a press conference that the law he signed last week "does not give anyone a license to discriminate."
But amid threats of crippling boycotts, the state's largest newspaper demanded in a front-page editorial that Pence "fix this now" by taking immediate action.
"We've got a perception problem," the embattled Republican governor acknowledged.
"After much reflection, and in consultation with leadership of the general assembly, I've come to the conclusion that it would be helpful to move legislation this week to make it clear that this law does not give businesses a right to deny services to anyone," Pence said.
Although he insisted he wanted to act this week and "move forward," Pence did not specify what remedy would be introduced and he stopped short of saying the law will be repealed.
Democratic state Senator Tim Lanane said Pence must take pro-active steps. "He can't just tinker with this language," Lanane told CNN.
Indiana, where same-sex marriage has been legally recognized since last October, has faced a backlash since the legislature passed the Religious Freedom Restoration Act.
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