Republicans aim to hold on to their House majority in November's midterm elections. Democrats need to gain 23 seats to take control of the chamber.
Spokespeople for both McCarthy and Ryan did not immediately respond to requests for comment.
Both men have opposed an effort by centrist House Republicans to force a vote on immigration. Twenty House Republicans have signed a so-called discharge petition, which would require a series of immigration votes if several more Republicans and all Democrats get behind it. Democrats are expected to support the effort.
Ryan and McCarthy met with President Donald Trump this week, and Ryan floated the idea of reopening negotiations on an immigration deal, Politico reported, citing several sources briefed on the meeting. Bipartisan negotiations flamed out earlier this year as Trump and conservative Republicans could not find common ground with Democrats and moderate GOP lawmakers on several demands. Democrats accused Trump of shifting in his demands throughout the talks.
Trump did not appear interested in starting up talks again when he met with Ryan and McCarthy, according to Politico.
Republican leaders are telling lawmakers who support the effort to force an immigration vote that they will bring immigration bills to the House floor next month, Politico separately reported.
Read the full Politico report here.