"I expect the Rules Committee to require a level of support that would leave only two candidates on the ballot at the convention," a senior Cruz Campaign aide told msnbc.
That committee, which writes the rules governing an open convention, is made up of 112 Republican delegates from around the country. Operatives for Cruz and Trump say they will have major sway over what the committee does.
"The Cruz people and Trump people are fighting hard to make sure their hard-core delegates get on the committee," said Barry Bennett, a Trump adviser.
Bennett said a voting bloc favoring the two leading candidates will run the committee.
"We'll be successful getting at least a majority — or supermajority," he projected, counting Trump and Cruz delegates together.
Both campaigns are backing a rule that would require candidates to achieve a minimum amount of support to get on the ballot, which could block Kasich in Cleveland. That would effectively end his campaign.
Kasich cannot mathematically win the nomination through the primaries, so his only hope is to win on a convention ballot.
"The committee is not going to oppose" a rule that the two leading campaigns support, Bennett added.