"We are working on potential hearings, potential depositions, I expect subpoenas are going to go out," he said. "We will move as expeditiously as possible, but we have to see what witnesses are going to make themselves available and what witnesses are going to require compulsion."
White House stonewalling, he added, "would just strengthen the case for obstruction."
The committee, now taking the lead on impeachment at Speaker Pelosi's direction, has told members to prepare to return to Washington next week for possible hearings. In a a letter to colleagues on Friday, Schiff said the committee would meet with Michael Atkinson, the Intelligence Community inspector general, behind closed doors.
On Friday, the House Foreign Affairs Committee, working with the Intelligence and Oversight Committees, subpoenaed the State Department for documents and information related to Trump's interactions with Ukraine and his phone call with that nation's president. They have also scheduled depositions with State Department officials, including some mentioned in the whistleblower complaint.
"The subpoenaed documents shall be part of the impeachment inquiry and shared among the Committees. Your failure or refusal to comply with the subpoena shall constitute evidence of obstruction of the House’s impeachment inquiry," Reps. Schiff, Eliot Engel, D-N.Y., and Elijah Cummings, D-Md., wrote.
Democrats, Schiff and other members have said they also want to hear from the whistleblower.
Democrats are also weighing reaching out to Rudy Giuliani, former DNI Dan Coats and his principal deputy, Sue Gordon, along with Attorney General Bill Barr, in their investigation into the Ukraine call and complaint.
Separately, Reps. Nita Lowey, D-N.Y., and John Yarmuth, D-Ky., the chairs of the House Appropriations and Budget Committees, respectively, wrote to the Office of Management and Budget on Friday demanding information about the withholding of military aid to Ukraine, which Trump had directed ahead of his July phone call with Ukrainian President Volodymyr Zelenskiy.
Pelosi and top Democrats hope to keep their impeachment inquiry narrowly focused on the Ukraine matter. Some want to prepare articles of impeachment by Thanksgiving and send them to the full House of Representatives by December, allowing the House to hold a floor vote on an impeachment before the new year.
"I think it's everyone's intention here to get this done in 2019," said Rep. Jamie Raskin, D-Mass., a member of the House Judiciary Committee.
Asked about the timeline, senior Democratic aides told ABC News the situation remained fluid, and was contingent on the committees' ongoing investigative work and efforts to follow up on the complaint, but pointed to Pelosi's repeated comments on working "expeditiously."
In a "Dear Colleague" letter to Democrats on Friday, Pelosi again said the impeachment probe will be "centered" in the Intelligence Committee, noting that the members of that panel will be working over the recess. She added that the Democratic caucus will hold a number of caucus calls next week to stay in touch.
Democrats, their oversight efforts stymied by the Trump administration in court at every turn, were wary of getting bogged down in legal disputes for documents and testimony that could further complicate and delay impeachment proceedings.
"Keeping it simple avoids court," said Rep. Eric Swalwell, D-Calif., a member of the Judiciary and Intelligence Committees. "Keeping it simple means ... we're not looking at empty chairs or having witnesses shouting at you like Corey Lewandowski," a reference to Trump's former campaign manager who appeared under subpoena before the Judiciary Committee in a raucous hearing earlier this month.