Oliver met with new Demons coach Steven King on Thursday and was informed he does not fit into the club's midfield plans next season.
Melbourne officials are happy for the four-time best-and-fairest winner to gauge interest among rival clubs.
Fellow 2021 premiership stars Christian Petracca and Steven May, as well as young defender Judd McVee, also appear set to switch clubs in the off-season.
Oliver, 28, is contracted for the next five seasons on a deal that would earn him more than $1 million a year.
The Demons would likely pay a significant chunk of that salary to get Oliver off their books.
"Today was a good chance to sit down with Clayton and share how Steven and the coaches see the game being played, and where the opportunities lie for him," Melbourne list manager Tim Lamb said on Thursday in a statement.
"We were up-front with Clayton around the role he is likely to play next season and the need for him to develop his craft beyond the midfield to have his biggest impact within the team.
"It was honest and respectful, and Clayton is now clear on what we see his role being next year, and how we think he can best contribute.
"Clayton is contracted for another five years. If he wants to explore his options in the upcoming AFL trade period, we will respect that.
"Should he remain in the red and blue, we will of course continue to support and embrace him as part of our club."
Oliver wanted to join Geelong last year, but a deal was not struck in the trade period.
The player was reportedly keen to continue at Melbourne under King, who has replaced Simon Goodwin after the 2021 premiership mentor was axed in August.
But being pushed out of the midfield could have led to Oliver spending more time playing in the Demons' reserves team.
Oliver hit out after the news broke on Thursday, posting on social media a picture of him playing for Melbourne's VFL affiliate Casey with the caption: "Actually doesn't look too bad on me."
He later flew to Sydney for a planned holiday, but not before delivering an expletive-laden blast at a Channel Seven journalist who attempted a door-stop interview with the player at the airport.
Petracca also considered leaving Melbourne last year and is again weighing his options despite being contracted for another four seasons, having met with Adelaide and Gold Coast this week.
King insists the Demons are keen to retain the four-time All-Australian and two-time best-and-fairest award winner.
"I've spoken to Christian on the phone," King told SEN radio on Thursday.
"As a coach, I'd love the opportunity to work with Christian and coach him.
"I fully understand the landscape we're in. I respect Christian's decisions now to explore his options, but he has got four years left on a contract.
"So it would have to be an amazing deal for our footy club for him to move. If things don't work out for Christian, it's a fresh start with me and Paul (Guerra) as CEO.
"I've got his back, around walking back into the club and playing great footy, and I'll back him in."
King added Petracca would be "pivotal" to his plans for next season and beyond.
Two-time All-Australian defender May, 33, has been told to look elsewhere, despite having one year left on his contract.
West Australian product McVee wants to return to his home state and is keen to join Fremantle.
The 22-year-old has played 65 games for Melbourne over three seasons.
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