Hagan supports the health-care law but she is taking the unusual step of spending money on advertisements designed to appeal to Republican voters who are skeptical of the measure. The maneuver is apparently intended to undermine enthusiasm in the GOP base for the Republican who is considered her strongest potential challenger in November.

In attacking a possible rival still embroiled in a primary contest, Hagan is embracing at least a variation of a tactic other vulnerable Democratic senators have used successfully in recent elections. Senate Majority Leader Harry M. Reid (Nev.) and Sen. Claire McCaskill (Mo.), who initially faced difficult reelection odds in 2010 and 2012, respectively, found ways to reach into GOP primaries and help weaker candidates emerge as their challengers.

If Tillis, 53, does not exceed 40 percent of the vote in North Carolina’s Republican primary Tuesday, he will face a potentially costly and divisive runoff in July. Polls and interviews with strategists on both sides of the aisle suggest that Tillis is leading but could be close to hitting the 40 percent threshold. His rivals include a Baptist minister with support from former Arkansas governor Mike Huckabee and a tea party favorite who appeared at a rally Monday with Sen. Rand Paul (R-Ky.).