The move comes after trade talks this week with the New York Giants fell through, sources told ESPN's Adam Schefter. Monroe, now a free agent, is likely to receive interest from the Giants, as well as the San Diego Chargers and Seattle Seahawks.
The Ravens had kept Monroe off the practice field for this week's mandatory minicamp because of a potential trade.
On Tuesday, coach John Harbaugh explained Monroe's absence as a "front-office-type situation."
"My understanding right now is that teams are inquiring about Eugene," Harbaugh said. "And when you're in that kind of a situation and there is a possibility of those kinds of things happening, you're pretty much obligated to pull back and not practice a guy."
Monroe hasn't practiced this offseason after undergoing season-ending shoulder surgery last season, but he announced a week ago that he was medically cleared to practice.
Monroe has been campaigning vigorously for the NFL to take medical marijuana off its banned substances list. He tweeted Friday that the Ravens were distancing themselves from him and his cause.
The move creates $6.5 million in cap room for the Ravens, but they take on $2.2 million in dead money in 2016 and $4.4 million next year. Ronnie Stanley, the No. 6 overall pick in this year's draft, has been working at Monroe's left tackle spot.
Information from ESPN's Jamison Hensley was used in this report.