After Spicer's resignation, Priebus said he supports Scaramucci "100 percent," according to news reports.
Scaramucci addressed the reported tension at the press briefing, saying he and Priebus have been personal friends for six years.
"We are a little bit like brothers where we rough each other up once in awhile, which is totally normal for brothers," Scaramucci said. "There's a lot of people in here that have brothers, so you get that. But he's a dear friend."
Scaramucci said Priebus was the first person he called Friday morning and the two had met beforehand. Scaramucci said the two are committed to getting the administration's message out.
"Listen, you guys are going to be very, very surprised about the relationship that I have with Reince and the closeness that we're going to have in terms of working to serve the president," he said.
Spicer's departure came six months into the Trump administration. His daily press briefings, often featuring tense exchanges with reporters, were frequently carried live on multiple networks. He was often ridiculed on NBC's "Saturday Night Live."
The tone of his tenure was set on his first full day as press secretary. At a Jan. 21 news conference, he verbally attacked the media and refused to take questions. He claimed they had spread "deliberately false reporting," including understating the size of the crowd at Trump's inauguration the previous day. He insisted that the crowd was "the largest audience to ever witness an inauguration — period, both in person and around the globe."
Spicer had taken over responsibilities as White House communications director after Mike Dubke resigned from the position in May. Reports swirled for months that Spicer would resign or be ousted.
Scaramucci is a former hedge fund star who has hosted the annual SkyBridge Alternatives (SALT) Conference. He is currently the U.S. Export-Import Bank's senior vice president.
Earlier Friday, a source close to the White House told NBC that Scaramucci met with Trump. In the meeting, Scaramucci was offered the role of communications director and accepted it, according to multiple reports.
When NBC asked one of its sources whether Trump would change his mind, the person said the president's mind was made up.
Priebus and top adviser Steve Bannon had resisted the appointment, NBC reported before Spicer's resignation became known. It said the two were kept out of the loop on the decision.
Sources within and outside the White House said Bannon had told Scaramucci he would get the job "over my dead body," according to NBC.
Scaramucci is a former CNBC contributor