The network is still debating just how far into the Republican primary process to venture. Most likely, the Kochs and their affiliated groups will opt for an approach similar to the one they took in 2014; they will largely stay out of specific races and instead work to shape the Republican landscape to ensure that their core issues — fighting the Affordable Care Act, rolling back various energy and environmental regulations, and overhauling government spending and the tax code — rise to the top of the candidates’ agendas.

But it is possible that one or both of the brothers, or a senior official of one of the affiliated groups, would individually endorse candidates, supporting them with money and events. Another option includes using issue-advocacy groups to support candidates they like and attack those they dislike.

Any audience with the Koch network provides Republican politicians with access to a valuable world. Candidates get the opportunity to pitch to the megadonors who can help them survive a grueling primary cycle. And even a subtle nod from the Koch brothers and their network — a high-profile speaking slot at an Americans for Prosperity summit meeting, or an invitation to the brothers’ biannual seminars — confers upon prospective candidates a stamp of approval that begets donor support.

At an Americans for Prosperity summit meeting in Dallas in August, both Mr. Paul and Mike Pence, the governor of Indiana, asked for and received a private meeting with David Koch. Mr. Christie was unable to make the gathering, but a month later he attended a private dinner with donors at David’s duplex on the Upper East Side of Manhattan.

“Whether we like it or not, one of the first primaries in either party is the fund-raising primary, and having access to that network helps you,” said Greg Mueller, a Republican strategist. “Being before a Koch-like network of prominent conservatives, who have financial resources, helps put you on the path to clear those early primary hurdles.”

In a speech Thursday, Tim Phillips, president of Americans for Prosperity, outlined what he described as the group’s “ambitious, steady effort” for the coming election cycle, complete with a “substantial” financial investment. “We’re going to be there throughout this year and the next year in 2016 and beyond,” Mr. Phillips said.

In many ways, what has made the network so powerful is its evolution into an almost shadow version of the Republican Party, including a top-shelf data and analytics firm, niche organizations targeting groups like Latinos and veterans, and grass-roots activists and staff members around the country.