WASHINGTON—A political cease-fire achieved by a further FBI investigation into allegations of sexual misconduct against Judge Brett Kavanaugh evaporated over the weekend, as the White House fended off accusations it had placed overly restrictive limitations on the probe of its Supreme Court nominee.

The one-week-at-most inquiry by the Federal Bureau of Investigation, brokered as a last-minute deal Friday between Republican Sen. Jeff Flake and Democrats on the Senate Judiciary Committee, was intended to satisfy concerns that allegations against Judge Kavanaugh weren’t being fully vetted before the full Senate took up his nomination.

But early signs that the FBI probe would be on a short leash inflamed Democratic criticism that President Trump and fellow Republicans weren’t out to explore fully the allegations, while the White House, Senate and FBI all appeared to shift responsibility for the scope of the probe elsewhere.

“The FBI’s hands must not be tied in this investigation,” Sen. Dianne Feinstein of California, the top Democrat on the Judiciary panel, wrote on Twitter. Later Sunday, Ms. Feinstein asked White House counsel Don McGahn and the director of the FBI to release a copy of the directive sent by the White House to the bureau outlining the scope of the investigation.

Late Saturday, the president had declared on Twitter that he wanted the FBI “to interview whoever they deem appropriate, at their discretion.”