The party lacks the establishment-anointed candidate that has led it into every presidential cycle for decades. Mitt Romney, a former Massachusetts governor, may come closest to that role, but he has kept a relatively low profile to avoid becoming a target. Newt Gingrich, a former House speaker, is well known but has political and personal liabilities. Newer faces like former Gov. Tim Pawlenty of Minnesota are barely known to most voters.

There is no guarantee that any Republicans on the sidelines would energize the party, but they seem to be getting almost as much attention as those clearly in the race. Reince Priebus, the Republican National Committee chairman, said last week that the race was still unformed and that he expected others to join.

The loudest and most persistent entreaties are directed toward Mr. Daniels, who for two terms as governor has shown that fiscal conservatism and political popularity can go hand in hand.

Mr. Daniels, who says Republicans should be more focused on addressing the country’s fiscal condition, worked in the Reagan administration and as budget director for President George W. Bush. An alumni network of those administrations, ranging from top contributors to field operatives, has elevated a whisper campaign into a forceful effort to enlist him.

The decision last week by another establishment favorite, Gov. Haley Barbour of Mississippi, not to pursue a presidential bid has increased pressure on Mr. Daniels, who drew national attention on Friday for saying he would sign a bill that cuts off Medicaid financing for Planned Parenthood in Indiana. His allies have started mapping out a campaign structure that could be fully in place this month, including a campaign headquarters in downtown Indianapolis.

“What sets Mitch apart from the other candidates who are currently running, he is very, very direct and very open about what needs to be done,” said Al Hubbard, a director of the National Economic Council. “I’m disappointed that the other candidates who are currently talking about running are reluctant to do that.”