McRaven is expected to accept the job, the sources said.
The tough-minded leadership and consensus-building abilities of McRaven, 58, were ultimately the deciding factors in the UT regents board's selection of McRaven, one source said.
But another high-ranking source said no deal had yet been completed with McRaven.
Richard Fisher, president and CEO of the Dallas Federal Reserve Bank, was also a finalist for the UT chancellor's job, the sources said.
McRaven would take over as chancellor after a tumultuous three years in which school president Bill Powers, Cigarroa and Texas regents could never seem to get on the same page. It resulted in seemingly constant acrimony.
Cigarroa announced in February he planned to return to surgery as head of Pediatric Transplant Surgery at the UT Health and Sciences Center in San Antonio when UT regents found his replacement.
"The leadership of Admiral McRaven is so universally respected, it's what this great university needs right now," one source told HornsDigest.com.
McRaven attended Texas on a track scholarship before graduating in 1977 with a bachelor's degree in journalism.
A member of the Naval Reserve Officers Training Corps, McRaven earned a master's degree from the Naval Postgraduate School and went on to command every level of the U.S. special operations, including the Navy SEALS.
McRaven is credited with organizing and executing Operation Neptune Spear, the raid on Osama Bin Laden that ultimately led to Bin Laden's death.
McRaven delivered the 2014 commencement address at Texas, which has been viewed on YouTube more than 2.1 million times.
Sources: Adm. McRaven offered Chancellor job
Horns Digest Top Stories
The Stars Are Out In AustinTom Herman will host some of the nation's top 2018 and 2019 prospects for the Texas Longhorns "Stars at Night" camp.
Horns Digest1:00 PM
Live Updates From Huge Recruiting WeekendHD is live in Austin for the Longhorns' huge recruiting weekend, which is expected to bring in several of the nation's top prospects.
Horns Digest9:56 AM
TEXAS WOMEN’S AD PLONSKY: ASSET OR DIVISIVE?Texas is the only school in Division I, II or III with a men's and women's athletic director. Longhorns' women's AD - Chris Plonsky - has a hall-of-fame resume, but her critics at…
Horns DigestYesterday at 5:22 AM
Should Texas vs. Texas A&M Game Return?HD POLL: Should Texas vs. Texas A&M rivalry game return to college football?
Horns DigestWednesday at 12:23 PM