Roma had taken a 2-1 advantage to Porto for the second leg of their last-16 tie on Wednesday but fell 4-3 on aggregate after a controversial VAR-awarded penalty deep into extra time.
There was no immediate comment from Roma on the future of coach Eusebio Di Francesco, who was already under pressure after a 3-0 derby defeat to Lazio on Saturday.
"On behalf of myself and everyone at AS Roma, I'd like to thank Eusebio for his work and his commitment", team president Jim Pallotta said in a release.
"Since returning to the club, Eusebio has always acted professionally and put the club's needs ahead of his own".
Roma is taking a "What have you done for me lately?" approach to its managerial position after losing to Porto in the UEFA Champions League Round of 16.
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Roma want a coach of experience to take over as they plan bigger changes in the summer with sporting director Monchi expected to leave for Arsenal.
Pallotta posted to social media: "Last year we asked for VAR in the Champions League because we got screwed in the semi-final and tonight, they've got VAR and we still get robbed".
Porto captain Hector Herrera, who was booked and will miss the quarter-final first leg, tested Robin Olsen with a bending effort just before the break when Roma lost skipper De Rossi following a crunching tackle by Danilo.
And a former Premier League champion may be first in line to replace him.
Ranieri was sacked by Fulham last month as the Premier League outfit labors in the relegation zone, and served in the same position at Nantes after leading Leicester City to its wonderful PL title.