The Obama administration says it is confident that Russia is responsible for the hacking of the Democratic National Committee and accuses Moscow of trying to interfere with U.S. elections.
All 858 people had previously been ordered removed from the country. The Department of Homeland Security's Office of Inspector General says fingerprint records are to blame.
Rep. Michael McCaul, a Texas Republican who chairs the House Homeland Security Committee, says he was apprised of the threat during a classified meeting.
The vote is a victory for President Obama as Republicans had wanted to strip funding for the president's executive actions on immigration from the bill.
President Obama late Friday signed a stopgap measure to keep the department running for another week, but the tussle over his executive action on immigration, linked to the funding, is not over yet.
The Senate approved a bill that funded Homeland Security through Sept. 30. The Republican leadership in the House decided to move forward with a more limited bill, but that failed.
The Department of Homeland Security, an agency repeatedly criticized for internal mismanagement and bloat, is the cornerstone of the new White House initiative to fight cybercrime.
The Speaker of the House's remarks, made on Fox News Sunday, come after days of back and forth over the measure, which also includes restrictions on President Obama's immigration initiative.