Former Massachusetts Governor Mitt Romney still leads the race for the Republican nomination -- but just barely, according to a Fox News poll released Thursday.

Romney’s support among GOP primary voters has dropped 6 percentage points in recent weeks, from 23 percent in early June to 17 percent in the new poll, which asked about announced and potential candidates.

Close behind Romney is undeclared candidate Texas Governor Rick Perry at 14 percent. Minnesota Rep. Michele Bachmann comes in at 10 percent -- up from 4 percent in early June.

Former New York City Mayor Rudy Giuliani, former Alaska Governor Sarah Palin and Texas Rep. Ron Paul all receive 9 percent. Giuliani and Palin have not declared their candidacy.

Businessman Herman Cain comes in at 5 percent. All others receive less than 5 percent.

Primary voters were also asked about their second choice candidate. If Giuliani doesn’t run, his backers mainly would go to Romney and the shape of the race would look about the same. If Perry decides against running, his supporters for the most part would split between Bachmann and Romney. And Palin backers go many different directions, though mainly split between Romney, Paul and Perry.

The Race Among Announced Candidates

The poll also asked Republican primary voters their preference among just the announced candidates.

When the field is narrowed, Romney’s support increases to 26 percent. He’s followed by Bachmann at 15 percent, Paul at 10 percent, and former House Speaker Newt Gingrich and Cain at 9 percent. Everyone else receives the backing of less than 5 percent.

Hypothetical Matchups

While President Obama tops each of the Republican candidates tested in hypothetical head-to-heads, he never gets to the level of 50 percent support.

Romney does best against the president and is the only GOP candidate to keep Obama’s edge to single digits. Obama tops Romney by 6 percentage points. In addition, Romney is the only Republican in these early matchups to top Obama among independents -- by 6 points.

Independents were essential to Obama’s victory in 2008 -- backing him 52 percent to Republican John McCain’s 44 percent (Fox News exit poll).

Obama has a 10-point advantage over both Pawlenty and Perry, and tops Bachmann by 11 points. The president’s widest gap -- 13 points -- is over Cain.

Finally, the poll shows women voters are neither more nor are they less likely to back Bachmann than any of the other Republicans over Obama.

The Fox News poll is based on landline and cell phone interviews with 904 randomly chosen registered voters nationwide and was conducted under the joint direction of Anderson Robbins Research (D) and Shaw & Company Research (R) from July 17 to July 19. For the total sample, it has a margin of sampling error of plus or minus 3 percentage points.

Sarah Palin is a Fox News contributor.