By Steve Holland and Lisa Lambert

WASHINGTON (Reuters) -- Jeb Bush, who has struggled to connect with voters in his bid for the Republican presidential nomination, raised $13.4 million in the latest quarter, his campaign reported on Thursday, an amount that pales in comparison to the heady, early days of his campaign.

The donations for the quarter, which included the month of August, when much political activity slows for summer vacations, put Bush behind only retired neurosurgeon Ben Carson in money raised among the 15 Republicans seeking their party's nomination for the November 2016 election. Carson's campaign raised $20 million, almost entirely from small donations. The leader in opinion polls among Republicans, Donald Trump, is largely self-financing.

Bush's campaign said he has $10.3 million in cash on hand.

His fundraising for the entire third quarter is only moderately more than Bush raised in the 16 days after he launched his campaign in July, when he pulled in $11.4 million.

In addition, Bush released the names of hundreds of people who have bundled together donations totaling $17,600 or more for his campaign.

While the Bush camp called it a landmark showing of transparency, campaign watchdog groups complained that releasing names at the $17,600 level made it hard to divine who were the mega-donors responsible for raising $100,000 or more.

Also on Thursday, Bush issued details of his health status, with his doctor reporting that at age 62, he is in excellent physical condition, takes a cholesterol medicine and has used exercise and careful eating habits to bring down his blood pressure and weight.

Other fund-raising totals reported by Republican candidates on Thursday - the deadline for campaigns to report donation totals to the Federal Election Commission - included $4.2 million brought in by New Jersey Governor Chris Christie, $4.4 million by Ohio Governor John Kasich and about $580,000 by Louisiana Governor Bobby Jindal.

Trump's campaign said he raised $3.9 million in the third quarter and has $254,772 in cash on hand. "Mr. Trump made a contribution of $100,779 during this quarter and has personally spent $1,909,576 since launching his campaign," his campaign said.

The dip in the pace of donations for Bush reflected his decline in polls of Republican voters who so far are favoring political "outsider" candidates such as Trump and Carson.

Bush campaign manager Danny Diaz said in a memo to Bush donors, some of whom have been rattled by Bush's decline in support, that the former Florida governor remains in a good position to challenge for the nomination.

“We knew from the start this was going to be a hard-fought and close race, but few could have anticipated just how volatile this field would be,” Diaz said.

The $13.4 million raised by Bush in the third quarter compared with $14.2 million raised in the same period in 2011 by the eventual 2012 Republican nominee, Mitt Romney, the Bush campaign said.

Senator Marco Rubio, a Bush rival for establishment Republican support and also from Florida, raised $6 million in the same period while reporting having $11 million cash on hand.

The Super PAC that is supporting Bush, Right to Rise, reported in July that it had raised more than $100 million. Unlike the candidates themselves, the Super PACs did not face the Thursday deadline for reporting donation totals to the Federal Election Commission. Sources have reported slowing donations to Right to Rise.

The Bush campaign has been working hard to build get-out-the-vote organizations in the early voting states of Iowa, New Hampshire and South Carolina and elsewhere.

"The overall effort supporting Jeb will be better funded than any other in the entire field," Diaz said.

(Reporting by Steve Holland and Lisa Lambert; Editing by Christian Plumb, Leslie Adler and Ken Wills)