The survey, from Democratic firm Public Policy Polling, gives Paul 16 percent support. Former Florida Gov. Jeb Bush, New Jersey Gov. Chris Christie and Rep. Paul RyanPaul RyanThe Hill Interview: Budget Chair Black sticks around for now Gun proposal picks up GOP support GOP lawmaker Tim Murphy to retire at end of term MORE (Wis.) all take 13 percent support, while Sen. Ted CruzRafael (Ted) Edward CruzWhatever you think the Alabama special election means, you’re probably wrong This week: Congress gets ball rolling on tax reform Week ahead: Senators work toward deal to fix ObamaCare markets MORE (Texas) takes 12 percent, Rubio takes 10 percent, and former Pennsylvania Sen. Rick Santorum and Louisiana Gov. Bobby Jindal each take 4 percent. New Mexico Gov. Susana Martinez gets 2 percent support.

Rubio dropped 6 points since PPP last polled nationwide, in May. He's faced a difficult few months as negotiations over immigration reform, on which he was a lead actor, brought him front and center in the national spotlight and drew the ire of conservative opponents.

Indeed, Rubio's support among voters who consider themselves very conservative slipped 9 points over that period.

And Cruz may be making up that difference. He's now drawing double-digit support, after taking only 7 percent of the vote in May. Much of that increase in support comes from very conservative voters, among whom he actually leads the field with 20 percent, more than even Paul takes.

On the Democratic side, former Secretary of State Hillary ClintonHillary Diane Rodham ClintonChris Murphy’s profile rises with gun tragedies DNC, RNC step up cyber protections Gun proposal picks up GOP support MORE continues to lead the pack, though her support has dropped since the May poll. She takes 52 percent of the vote, down from 63 percent in May, while Vice President Biden takes 12 percent, virtually unchanged from the last poll. No other Democratic candidate cracks double-digits.

Though Clinton still leads every potential Republican challenger, they've managed to close her margins over the past few months.

Her lead on Christie is within the margin of error, with only 43 percent to 42 percent. In May, Clinton led him by 3 points.

Ryan takes 44 percent to Clinton's 46 percent, and Bush trails her by only 3 points, with 41 percent to Clinton's 44 percent support. All three take at least 10 percent of Democrats and a significant portion of independents.

That's why Paul fares worse against Clinton. Buoyed only by solid support in his own party, he trails her by 8 points, with 39 percent support to Clinton's 47 percent.

The survey was conducted among 800 registered voters, including 500 Republican and 418 Democratic primary voters, July 19-21, and has a margin of error of plus or minus 3.5 percentage points overall. For the Republican portion, the margin of error is plus or minus 4.3 points, and for the Democratic portion, the margin is plus or minus 4.7 points.