Republican U.S. Representatives Ralph Norman and Mark Meadows, prominent members of the conservative Freedom Caucus, told reporters they did not think either of the bills had enough support to pass the Republican-controlled House.
The Trump administration earlier this year began enforcing a policy that separated members of families that cross U.S. borders illegally.
Meanwhile, three of the 12 immigrant children at a Catholic Charities shelter in Fort Worth, near Dallas, were expected to be reunited with their family on Thursday.
In a rare public reversal, President Trump says the US will allow undocumented families to stay together, even with his "zero tolerance" border policy in place. "Schumer used to want Border security - now he'll take Crime!" That bill never got a vote in the House.
Despite Trump's executive order, a host of unanswered questions remain, including what will happens to the children who were already separated from their parents and where the government will house all the newly detained migrants, with the system already bursting at the seams.
Lawyers filed a memorandum to a settlement in California that governs how children are handled when they are caught crossing the USA border illegally. "I was not put in a freezing cell".
The House prepared to vote on a Republican immigration bill containing language aimed at halting the taking of immigrant children from parents being detained for illegally entering the US.
The other was a compromise between GOP moderates and the party's conservatives that included an opportunity for citizenship for the young immigrants.
Under Trump's new executive order, migrant children will be kept with their families where possible, though they will remain in detention facilities.
Democrats oppose both measures as harsh. Senate Minority Leader Chuck Schumer, D-N.Y., told Trump he could change the situation with the stroke of a pen. "It may be a compromise with the devil, but it is not a compromise with the Democrats".
The fact that votes on immigration bills are happening at all is because of an effort by moderate Republicans who grew exhausted of their party's failure to act on immigration.
King said that if the House fails to pass an immigration bill attention would shift to see what the Senate could do. "Republicans must get rid of the stupid Filibuster Rule-it is killing you!" "This immediate crisis seems to be going away".
Republicans joined Democrats to vote by 193-231 to scupper the more hawkish bill. The Senate, where Republicans hold a slim 51-49 majority and passing most legislation requires 60 votes, is working on its own bipartisan version of the legislation.
For days, Trump and his top administration officials were unwilling to unilaterally reverse the separation policy, insisting that congressional action was required. Ryan also complained that Congress is dysfunctional without deadlines, and after the Supreme Court earlier this year essentially delayed the need for a vote on a DACA resolution until after the midterm elections, there was no urgency to act on Capitol Hill.
But GOP leaders were eager to hold the votes anyway.
One bill promotes a hardline approach to immigration, while the other would give legal protections to young immigrants but Democrats say it would also cut legal immigration levels.
Passage of the GOP compromise was always a long shot, but failure may now come at a steeper price.
"We can enforce our immigration laws without breaking families apart", Ryan said Wednesday before Trump announced he would sign an executive order to end breaking up families.
"For those children still in Border Patrol custody, we are reuniting them with parents or legal guardians returned to Border Patrol custody following prosecution", a CBP spokesperson said in the statement. It did so by eliminating the 20-day cap on holding minors and allowing indefinite detentions. Family separation is wrong.
Moderate Republicans forced the immigration debate to the fore by threatening to use a rare procedure to demand a vote.
Trump campaigned on stopping illegal immigration and has fiercely defended his administration's actions.