WASHINGTON—Republicans forged ahead with a hearing Monday to consider sexual-assault allegations against Supreme Court nominee Brett Kavanaugh and set a deadline of Friday to determine whether his accuser appears, after days of political wrangling over how best to handle accusations that threaten his nomination.

White House Spokesman Raj Shah said the president won’t look at naming any replacement nominee unless there is a clear need. The administration, he said, was going “full steam ahead” to support Judge Kavanaugh, who has denied the assault accusations.

GOP leaders in the Senate said they were continuing to seek the testimony of Judge Kavanaugh’s accuser, California college professor Christine Blasey Ford. Earlier, Republicans rejected calls from her attorneys and Democrats for an investigation of the allegations as a condition for her appearing at the hearing.

Sen. Chuck Grassley (R., Iowa), chairman of the Judiciary Committee, said the hearing would start at 10 a.m. Monday, and he pressed Dr. Ford to testify. In a letter to her lawyers, he gave her a deadline of 10 a.m. Friday to submit her biography and prepared remarks if she planned to testify.

Dr. Ford’s attorneys issued a statement late Wednesday calling for more witnesses to be involved in the hearing—not just Judge Kavanaugh and Dr. Ford. It didn’t say whether Dr. Ford would attend the hearing. “The rush to a hearing is unnecessary, and contrary to the Committee discovering the truth,” the attorneys wrote.