WASHINGTON—An attorney for the California college professor who has accused Supreme Court nominee Brett Kavanaugh of sexually assaulting her when they were teenagers said she would be willing to testify next week before the Senate Judiciary Committee, an offer that could break a partisan logjam over whether the FBI should investigate her allegations.

Christine Blasey Ford isn’t willing, however, to go before the panel by Monday, when a hearing is now scheduled, her lawyer wrote in an email Thursday to committee staff members. “She wishes to testify, provided that we can agree on terms that are fair and which ensure her safety,” wrote the lawyer, Debra Katz, who added that Dr. Ford had been receiving death threats.

Before Thursday, Dr. Ford had said she wanted the Federal Bureau of Investigation to reopen a background investigation into Judge Kavanaugh before she testified, an idea rejected by Republicans. In Thursday’s message, Ms. Katz said that continues to be Dr. Ford’s “strong preference,” but it doesn’t appear to be an unconditional demand.

Ms. Katz spoke with Senate Judiciary Committee staff Thursday, but the two sides didn’t reach an agreement, congressional aides said. Ms. Katz raised questions about security and what media would be present. They also discussed other sticking points such as additional witnesses. Dr. Ford has asked for Judge Kavanaugh to testify first. Her camp is also pushing to hold the hearing on Thursday, according to congressional aides.

The conversations are expected to continue, the aides said. Ms. Katz cancelled two planned television appearances Thursday night, citing the effort to make progress with the committee.