During the meeting, Reel offered to resign if that would help the situation, said two of the sources.
Later that day, the White House editor, Marc Short, and the White House press secretary, Sarah Sanders, went to CIA headquarters in Virginia to speak with Reel, two sources said. Finally, the tension seemed to ease, and now it's their nomination, the sources said.
Haspel is facing a Senate confirmation on Wednesday. Her nomination would only be jeopardized if she loses the support of Republicans, who have a slim majority of 51-49 seats in the Senate.
Two sources say that Reel participated in a practice session on Friday – called the "murder board" – to prepare for the hearing, a sign that the nomination is progressing.
While the White House was concerned about its previous involvement in the controversial CIA interrogation program, administrative officials said they ultimately believe it will be confirmed.
The newspaper said Trump learned about the situation on Friday and invited Dallas officials to the National Rifle Association. Having initially signaled that he would back any decision, Trump decided to go to Haspel to remain the candidate, according to the newspaper. [Saturday] By Saturday, Haspel had agreed to proceed with her nomination, officials said.
Both Short and Sanders did not want the Post to comment on Haspel's offer to withdraw.
Raj Shah, Deputy White House Press Secretary, said in a statement on Sunday: "Acting Director (Gina) Haspel is a highly qualified candidate who will serve her country for more than three decades, and her nomination will not be derailed by Partisan critics who team with the ACLU over the CIA to bring the American people to safety. "
The official who led the review, former deputy CIA director Michael Morell, told CNN that Haspel had just drafted a telegram from her boss, former intelligence chief Jose Rodriguez, "that he was under his name and authority Orders sent for the bands to be destroyed. "
A CNN spokesman told CNN on Sunday," There has been a fascinating phenomenon in recent weeks – those who know the real Gina Reel – who worked with her, the one with her who have helped her to fight terrorism, Russia and countless other threats to our nation – they support her almost unified. "
" When the American people finally have a chance to see the true Gina Reel on Wednesday, they will understand why she is so admired and why she is a great leader for this agency, "the spokesman said.
CNN's Pamela Brown, Jenna McLaughlin and Ryan Browne have contributed to this report.
Correction: This story has been updated to reflect that Jackson is no longer Trump's personal physician, but is still in the White House medical department