Former Cruz delegate chief Ken Cuccinelli and his organization, Trusted Leadership PAC, are working to convince the convention Rules Committee that starts here on Thursday to consider proposals that would close primaries to only Republican voters and decentralize the control over the primary process away from the Republican National Committee, Cuccinelli's spokeswoman Mallory Rascher said.
Cruz is not publicly a part of the effort, and his campaign office confirmed he is not getting involved.
The effort is a response to Trump's successes this year in open primaries where non-Republicans were allowed to vote. The presumptive nominee consistently railed against the primary and caucus process, calling it "rigged."
Louisiana RNC Committeeman Ross Little Jr., who was a big supporter of Cruz and sits on the Convention Rules Committee, said he is committed to making the changes to close primaries.
"My focus is on for the 2020 convention, that we would have incentive for states to have closed primaries for states to elect Republicans and to increase the size of the total convention in four years," Little said.
The move may face opposition from RNC loyalists who see it as a power play by Cruz that could limit their ability to get a moderate or establishment favorite elected next cycle.
The attention to this proposal may also have the side effect of drawing momentum from the effort to unbind delegates and potentially keep Trump from winning the nomination on the first ballot.
Steve Duprey, the national RNC committeeman from New Hampshire, where independents play a large role in primaries for both major parties, told reporters he expects plenty of debate about it on the committee, but made a case for keeping primaries open if states want them that way.
"I think that's another spirited debate will have, there's arguments both ways," Duprey said. "Personally, I know in New Hampshire the fact that we have independents who come vote in a presidential primary then attracts people to vote for our candidates in the fall. ... In our state it makes sense, in some states it doesn't make sense. That's the beauty of how it works."