House Oversight Chairman Darrell Issa (R-Calif.), an outspoken critic of the legislation, said early Saturday morning that Majority Leader Eric Cantor (R-Va.) promised him the House will not vote on SOPA unless it has a consensus.
"While I remain concerned about Senate action on the Protect IP Act, I am confident that flawed legislation will not be taken up by this House," Issa said in a statement.
Smith did not mention Issa's claims in his statement vowing to push ahead on the bill.
Cantor's office did not respond to requests to comment on his commitment to shelve the bill.
SOPA and Protect IP are designed to go after foreign websites that offer illegal copies of music, movies and TV shows with impunity. The bills would empower the Justice Department and copyright holders to demand that search engines delete links to sites “dedicated” to copyright infringement. Ad networks and payment processors would be prohibited from doing business with the sites.
Movie studios, record labels and business groups say the law is necessary to curb online copyright infringement, which is hurting businesses and destroying jobs.
But consumer groups and Web companies warn the bills would stifle innovation and censor free speech. They say the legislation would impose an unreasonable burden on websites to police user-generated content and could lead to legitimate websites getting shut down.
On Saturday, the White House weighed in for the first time, expressing concern about the legislation's potential to restrict the openness of the Internet.
"Any effort to combat online piracy must guard against the risk of online censorship of lawful activity and must not inhibit innovation by our dynamic businesses large and small," the president's advisers wrote in a statement.
On Monday, White House spokesman Jay Carney said the president "will not support legislation that reduces freedom of expression."
But the administration has emphasized it is committed to passing a law this year to address foreign infringing websites.
In a major concession to the bill's critics, Smith said last Friday he will drop a controversial provision that would have required Internet service providers to block infringing websites through a process known as Domain Name System filtering. Sen. Patrick Leahy (D-Vt.), sponsor of the Senate bill, has also said he is willing to re-examine the provision.
The White House opposes the site-blocking provision.