It left open speculation as to whether the White House would attempt to push through a nomination without the Senate's approval, through a recess appointment.
Obama was thwarted in 2012 when he tried to appoint members to the National Labor Relations Board while the Senate was on a break, but holding pro forma sessions every three days. The Supreme Court eventually ruled that the Senate was not on an official recess then, as it is through the rest of this week.
When asked about the possibility of a recess appointment, Schultz pointed to the president's comments on Scalia's passing Saturday, when he said he would put a nominee forward "in due time."
"There will be plenty of time for me to do so and for the Senate to fulfill its responsibility to give that person a fair hearing and a timely vote," Obama said. "These are responsibilities that I take seriously, as should everyone. They are bigger than any one party."
The White House's acknowledgement that it will put a nominee in front of the Senate is the first step in what could develop into a tumultuous political fight in Obama's final year in office.
As of Sunday afternoon, Senate leadership has not replied to ABC News' requests for comment about whether it will hold confirmation hearings once a nominee is announced.