"I'm spending every opportunity I can on the House floor to whip up support," he said.
The resolution has sparked competing "Dear Colleague" letters from supporters and foes of the bill.
On Monday, Rep. Gerald Connolly (D-Virginia) and four other lawmakers asked that the measure be tabled, citing Turkish support of U.S. military efforts in Iraq and Afghanistan, including an American air base in Turkey.
"The United States cannot afford to damage the strategic relationship with Turkey — a key NATO ally currently deployed with U.S. forces overseas," the letter states.
On Tuesday, Reps. Brad Sherman (D-Sherman Oaks), Schiff and three other lawmakers fired off a letter demanding a vote, in which they argued that the U.S. had long been a global human rights leader, noting past congressional resolutions on the Holocaust and atrocities in Rwanda, Bosnia and elsewhere.
"On the issue of the Armenian Genocide, however, we lag behind," they said.
In recent days, State Department officials have said the Obama administration opposes the resolution.
But Schiff said that Turkey's standing among U.S. leaders has been eroded given its recent diplomatic clash with Israel and its opposition to economic sanctions against Iran.
Schiff said the administration's opposition to the genocide measure has been "pro forma."
"They have not been lobbying members and making the kind of calls we have seen in the past," Schiff said. "I think the Turkish resistance will not be as effective as it has been."
Armenian National Committee of America Executive Director Aram Hamparian said, "We are confident that a bipartisan majority, if given an opportunity, will pass this measure."
Van Krikorian, a spokesman for the Armenian Assembly of America, said that getting to a vote would be difficult, in part because the administration is planning a sizeable arms sale to Turkey.
Schiff said he will reintroduce the resolution if necessary next year, but that the odds will grow longer in the next Republican-controlled House.
"This week I think we have maybe our greatest opportunity yet, and our best opportunity for a long time," Schiff said.