The Florida Times-Union was first to report a deal had been struck on Friday.
An ESPN.com source said Monroe is expected on the practice field with the team Friday night. The team held its final public practices of training camp Thursday. Monroe missed 22 workouts over 12 days.
The Jaguars selected Monroe out of Virginia in an effort to refortify their offensive line, which was decimated by injuries during a 5-11 season in 2008.
Thomas is expected to be the starter on opening day in Indianapolis, but it wouldn't be surprising if the Jaguars eventually shift to Monroe to get their youth experience in what they've called a retooling season.
Monroe's deal was expected to be worth a little more than the reported five-year, $28.5 million ($18 million guaranteed) that the ninth pick, Green Bay Packers defensive tackle B.J. Raji , agreed to Thursday, according to the Times-Union, which cited an unnamed NFL source.
"It gave a better idea of where I should be," said Monroe, whose deal could be worth as much as $34.5 million with escalators.
Although he's not expected to play in Monday night's preseason opener at Miami, Monroe could be in the starting lineup by the end of the month.
"I feel he'll have a bright future as a Jaguar," general manager Gene Smith said. "He's a quick study. I know he'll come in and compete and do whatever it takes to get prepared to compete to be a starter. ... It was nice to see him get out there and compete against a couple of our defensive linemen.
"He'll catch up in a hurry."
Paul Kuharsky covers the AFC South for ESPN.com. The Associated Press contributed to this report.