The bill upholds the administration's earlier position of denying detainees access to courts to challenge their detention but placed more restrictions on interrogation techniques by the military and the CIA, The Washington Post said.
Bush issued a statement thanking the House and calling the existing system "one of our most effective tools in the war on terror."
The administration's existing system was struck down by the Supreme Court in June, as it ruled some Geneva Conventions rights were being denied.
Bush said he hoped the Senate would pass the bill before Congress recesses on the weekend.
The bill would pave the way for trials of more than 20 suspected terrorists, including alleged Sept. 11, 2001, planner Khalid Shaikh Mohammed, who is being held at the U.S. military prison at Guantanamo Bay, Cuba, The Los Angeles Times said.
Notable deaths of 2014 [PHOTOS]