But some of the collaboration is becoming more overt.

Over the last couple of months, network members filed amicus briefs accusing Mr. Trump of overstepping his authority on matters ranging from immigration to his administration’s efforts to block a merger between AT&T and Time Warner. And last month, the Senate Judiciary Committee passed a bill to protect the special counsel investigating Russian meddling in the 2016 election, for which network members had lobbied.

“It was a kumbaya moment,” said Lisa Gilbert, an official at the left-leaning watchdog group Public Citizen, which helped coordinate lobbying by groups across the political spectrum.

A coalition involving many of the same groups is working to quickly mobilize mass protests across the country if Mr. Trump acts to impede the special counsel investigation. And next month organizers of a group of leading donors and operatives from the right and left called Patriots and Pragmatists are expecting their biggest turnout yet at a meeting in San Francisco, according to people familiar with the planning.

The Niskanen Center hosts a semimonthly invitation-only gathering of Trump critics called the Meeting of the Concerned, which attendees are asked to keep confidential. While it is attended primarily by a rotating cast of Never Trump Republicans — including the pundit Mr. Kristol and the former Representative Mickey Edwards — meetings sometimes include a Democrat or two presenting research or analysis on relevant issues. Attendees have included Ian Bassin, a former Obama White House lawyer who founded a watchdog group, Protect Democracy, that has sued the Trump administration and that has brought on staff members and advisers — including Mr. Taylor — from conservative or Republican backgrounds.

Other Republicans have been wary to get involved, Mr. Bassin said. “There is a troubling dynamic happening where anytime a conservative expresses concerns, they get branded a Never Trumper and are excommunicated from the American right.”

But Mr. Edwards, a Republican representative from Oklahoma from 1977 to 1993, said he was not worried about stigma. “Party identity really is being set aside. We have bigger fish to fry,” he said. He has become involved in a number of cross-partisan coalitions challenging Mr. Trump in recent months, including working with the former Obama administration lawyers Neal Katyal and Joshua Geltzer to file a legal brief in March in a court case opposing Mr. Trump’s proposed restrictions on travel from predominantly Muslim countries. “It’s really a much more free-form kind of resistance. It’s a movement more than an organization.”