An early test looms Friday at the Conservative Political Action Conference in Washington, where Mr. Bush will make his first appearance before an audience of conservative activists since he began preparing for a 2016 campaign in December.

“He is the kind of candidate that a lot of Republican donors have been looking for—someone who can win the general election and reach out beyond the party’s core supporters,” said John Rowe, co-chairman of the Illinois Business Immigration Coalition, which supports allowing undocumented workers to earn citizenship. Mr. Rowe said he donated $25,000 to Mr. Bush’s Right to Rise political committee at last week’s fundraiser.

Mr. Bush is also scheduled to meet Friday with Tony Perkins, an influential social-conservative leader and the president of the Family Research Council, who declined to invite him to the group’s convention last year. At the time, Mr. Perkins said Mr. Bush and New Jersey Gov. Chris Christie , another possible presidential candidate, were “quite frankly not at the top of the list when it comes to values voters.”