Their support is public because they have given money to a Romney super PAC, which is required under federal law to disclose the gifts it receives. But there is a whole other swath of influential Romney supporters whose names voters can only guess at, because the campaign refuses to reveal them.

They are a category known as “bundlers,” people who write big checks themselves and press friends and associates to donate. They became increasingly influential after campaign finance reform in 2002, and now account for hundreds of millions of dollars that fuel national candidacies.

“In some ways bundlers are more important, or at least as important, as the donors themselves, because someone who is able to bundle half a million dollars together has influence,’’ said Richard Briffault, a professor at Columbia Law School.

By law, Romney and other candidates have to reveal only the names of bundlers who are lobbyists. But, Briffault said, “It goes toward the whole notion of transparency in the campaign finance system. Obviously, the money matters.”

If their candidates win, bundlers are often richly rewarded with ambassadorships and other administration posts, or personal access like overnight stays in the White House. According to the Center for Public Integrity, which tracks campaign finance issues, about 80 percent of Barack Obama’s and George W. Bush’s bundlers who raised $500,000 or more went on to win appointments.

Critics say that by failing to disclose the names of his bundlers, Romney is out of step with past Republican presidentidal candidates — and his own practice in 2008 — as well as that of his Democratic opponent, President Obama. Obama revealed his bundlers’ names in 2008 and is doing so again this election year.

The Romney campaign won’t say why it has declined to reveal names, despite pressure to do so from advocacy groups and the Obama camp.

Said Andrea Saul, a spokeswoman for the Romney campaign: “We disclose all of the information about our donors as required by law, and anyone who is interested can review it publicly.’’

Bundlers are often prominent supporters in cities across the country who hold house parties at which the candidate appears and attendees dig deeply into their pockets for the chance to rub elbows. The goal is to get lots of friends to give the $2,500 maximum donation to the candidate. Donors also can give up to $30,800 per year to a political party.

Super PAC donations have no limits and cannot be directly connected to the campaign.

President Obama’s bundlers in Boston are a who’s who of successful business people, from hedge fund executive Scott Nathan and retired advertising mogul Jack Connors to Bain Capital partner Jonathan Lavine and public relations veteran Larry Rasky. Each has raised more than $500,000 for Obama, according to the campaign’s disclosure and figures put together by the Center for Responsive Politics, a Washington group that tracks money in campaigns.

Clues about Romney’s bundlers, by contrast, must be gleaned from event invitations, eyewitness accounts, and news reports. For instance, Reebok’s Fireman and his wife, Phyllis, are hosting a fund-raiser at their home in Chestnut Hill on Sept. 28. As hosts, the Firemans have committed to raise $100,000, while “VIPs” will bundle $50,000, and $10,000 will fetch a photo with the candidate. Last month, billionaire Bill Koch threw a $50,000-a-head fund-raiser for Romney at his home in Osterville; others hosted events on Nantucket and Martha’s Vineyard.

Putnam Investments chief executive Robert Reynolds attended the Nantucket event but said he is not a bundler, just an individual supporting Romney, whom he considers a friend. He also has given $100,000 to the Romney super PAC. He said he is not troubled by the Romney camp’s lack of disclosure on bundlers.

“To me, bundlers is a very definitional word. What is a bundler? I think it’s hard to define,’’ Reynolds said. “If Obama’s having a reception here — he’s had a couple hosted here — and the person invited 20 people, is he a bundler?”