A news conference would follow on Monday, the source said.
During conversations about the job, Strong impressed Texas with his commitment to the total student-athlete, a source said. The Longhorns were also impressed by individuals both inside and outside of football who vouched for Strong, according to the source.
Strong met with his staff for about 15 minutes on Saturday morning and told them he had not decided to accept Texas' offer to become its coach.
Strong told Louisville athletic director Tom Jurich on Friday night that he had not yet accepted the job, Jurich told ESPN's Brett McMurphy. Strong wanted to talk to Jurich and Louisville president James Ramsey on Saturday, another source said. Jurich was in Colorado on vacation on Friday night and his return was delayed by weather.
Cardinals football spokesman Rocco Gasparro said Saturday that Strong told his staff that "no decision had been made" on whether to leave the school after four years, and there was no timetable on a decision.
"It's a difficult decision for him," Gasparro said.
Louisville defensive line coach Clint Hurtt, who was sanctioned in the NCAA's investigation of the University of Miami, is not expected to follow Strong to Texas.
Strong, 53, is coming off his second straight double-digit victory season. He took over a program that was coming off three straight non-winning seasons and has gone 37-15.
On Friday, two other coaches Texas reportedly had interest in -- Baylor's Art Briles and UCLA's Jim Mora -- said publicly they were staying in their current jobs.
Texas is looking to replace Mack Brown, who stepped down after four straight seasons that failed to live up to expectations, including this season's 8-5 record. New Texas athletic director Steve Patterson, hired in November to replace longtime AD DeLoss Dodds, interviewed Strong this week.
Strong had two stints as an assistant at Florida, first under Steve Spurrier and then as defensive coordinator for Urban Meyer's two national championship teams. In 2010, Louisville hired him to take over the program, and he quickly righted the Cardinals, who are moving to the Atlantic Coast Conference next season.
Strong would be Texas' first black head football coach and would inherit a program aching to return to its place among the nation's elite programs.
Brown's Longhorns won the 2005 season national championship and returned to the national championship game after the 2009 season. But the Longhorns fell to 5-7 in 2010 and have lost at least four games each of the past three seasons.
That drop-off, including an 18-17 mark in the Big 12 in the past four seasons, frustrated Texas fans, who demand much more from the wealthiest athletic program in the country, which sits in the middle of the most fertile high school recruiting grounds in the country.
Information from ESPN's Brett McMurphy and Joe Schad and The Associated Press was used in this report.