The emerging and comprehensive House plan, like the one in the Senate, has a proposed path toward citizenship.
Republicans Flake, McCain and Graham recently meet with a number of House Republicans to explain their efforts.
"I don't want to say what their positions were, but they were cordial," said Flake, elected to the Senate in November after 12 years in the House. "They listened."
Flake said he expected the Senate to pass a comprehensive bill while the House approved a limited one. But differences between the two measures could then be worked out, he said.
McCain said he believed "we can convince our House Republican friends - if we can make sure that they are convinced that we have an effective control of the border and it is not amnesty."
McCain said that while he recognized the political and economic advantages of immigration reform, there was a humanitarian need as well, which he said he knew well as an Arizonan.
"Young people have been brought across the border by a coyote (smuggler) and told, 'Tucson is over there,' and have died in the desert," McCain said.
All the members see their immigration effort as a test of whether Congress, when it is divided, can get anything of significance done.
"It could help get Democrats and Republicans to work together," McCain said. "That is what the American people want."
If members as far apart as Schumer and McCain can come together, it will send a message, said Graham.
"If these two guys can deal with an issue as complicated as immigration and wind up with a product good for the country, who knows, good government may break out."
"But," Graham added with a sigh, "I doubt it."
(Editing by Fred Barbash, Mary Milliken and Peter Cooney)