House Republicans failed to agree on immigration legislation June 7 as leaders pleaded with them to pursue a GOP-only solution instead of forcing action on bill that would draw votes from Democrats but a veto from President Donald Trump.

Lawmakers left a two hour meeting with House Speaker Paul Ryan that they hailed as a productive conversation even though there was no consensus on their next steps.

Ryan is pressing them to accept a bill that would follow the “four pillars” Trump laid out earlier this year, including a path to citizenship for some young immigrants, enhanced border security, eliminating...