Many Democrats have pushed for an immediate impeachment trial to hold Trump accountable and prevent him from holding future office, and the proceedings could still begin by Inauguration Day.
Psaki noted that during Trump's first impeachment trial last year, the Senate continued to hold hearings each day.
Pelosi told reporters on Friday that the nine House impeachment managers, who act as the prosecutors for the House, are working on taking the case to trial.
McConnell is open to considering impeachment, having told associates he is done with Trump, but he has not signaled how he would vote.
No president has ever been convicted in the Senate, and it would take a two-thirds vote against Trump, an extremely high hurdle.