The preference would be for Ford to testify next Thursday, and she doesn't want Kavanaugh in the same room, her attorney told Judiciary Committee staff in a 30-minute call that also touched on security concerns and others issues.
With his comment, Mr Trump went against advisers who had counselled him to stay out of the fray.
"With the battle over the confirmation of Judge Kavanaugh to the Supreme Court being charged by sexual assault accusations and at the nearly one-year anniversary of the start of #MeToo, you'd think that America would be split by gender on this", said Cliff Young, president of Ipsos.
But the White House would have to ask for it.
The president has indicated he wants Ford to testify, saying, "I really want to see her, to see what she has to say".
Before the president's tweets Friday, counselor to the president, Kellyanne Conway told reporters there was "no reason" to attack Ford. "But I don't think you can delay it any longer".
She asked Kavanaugh, "Since you became a legal adult, have you ever made any unwanted requests for sexual favors or committed any verbal or physical harassment or assault of a sexual nature?" as well as "Have you ever faced discipline or entered into a settlement related to this kind of conduct?"
Lisa Banks, one of Ford's lawyers, said in a statement to CNN on Wednesday that Ford and her family had received threats, which forced them to leave their home.
Both cited an email the lawyers sent to the committee seeking to negotiate a new date for the hearing, after committee chairman Chuck Grassley set a deadline of Friday morning to agree to appear on Monday.More news: Winds, rain intensify as Hurricane Florence pummels North Carolina
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She said she is willing to testify at earliest on Thursday.
Kavanaugh is scheduled to appear again before the Senate Judiciary Committee next Monday following allegations that have endangered his appointment to the Supreme Court.
Mr Trump's tweet could affect GOP support among women going into the midterm elections.
Asking Kavanaugh to go first is "not doable", according to the source, noting he would need to respond to any new accusation that comes up at the hearing.
Sara Fagen, another longtime friend who worked with Kavanaugh in the White House, told a story about how Kavanaugh helped her the day of the September 11, 2001 attacks. That response has contributed to him continuing to say Ford should come forward with her story.
There are no Republican women on the Senate Judiciary Committee, and the idea of an outside questioner being brought in has been floated publicly.
But Katz says that a Monday hearing "is not possible and the committee's insistence that it occur then is arbitrary in any event". "They've delayed it a week already".
A senior Senate Democratic aide noted that reopening Federal Bureau of Investigation background checks was fairly routine; 10 such probes into judicial nominees had been reopened in the last three months alone, the aide said. His Republican party accuses the Democrats of using Ford as a last-minute spoiler to try and wreck Kavanaugh's path to the highest court, where he would tilt the bench firmly to the right for possibly years to come.