Posted to Byrne's Twitter page on Monday was an entry that read "And the new Arizona football coach and his family is......."
The post included a link that, when clicked, led to a photo of Rodriguez with his family.
Byrne will officially introduce Rodriguez at a news conference Tuesday at the McKale Center, the school said. He replaces interim coach Tim Kish.
Arizona (3-8, 2-7 Pac-12) fired Mike Stoops last month after struggling to a 1-5 start.
Rodriguez coached Michigan from 2008-2010 and West Virginia from 2001-2007. Rodriguez posted a 60-26 record at West Virginia, finishing first in the Big East four times.
Rodriguez was 15-22 at Michigan, including 6-18 in Big Ten play, before being dismissed.
During Rodriguez's tenure, the school acknowledged that it was guilty of four NCAA violations. It was put on three years of probation, though Rodriguez and the school avoided major penalties in part because the NCAA agreed that the coach didn't fail to promote an atmosphere and compliance in his program.
Rodriguez has been working as an analyst for CBS Sports this year, but had made it clear he wanted to get back into coaching as soon as possible.
Byrne fired Stoops on Oct. 10, two days after the Wildcats lost to Oregon State. The team was 1-5 at the time, with the only victory over FCS member Northern Arizona, and 10 straight losses to FBS schools.
Stoops' contract ran through 2013 under an extension granted in
2008, and Byrne has said the buyout for the remainder of it will be
about $1.4 million.
Stoops was 27-38 in conference games.
Kish, the team's defensive coordinator, took over as interim coach and the team won two Pac-12 games, over UCLA and last Saturday 31-27 over rival Arizona State. The Wildcats conclude their season Saturday at home against Louisiana-Lafayette.
Information from ESPN college football reporter Joe Schad and The Associated Press was used in this report.