Roma have announced the sacking of Eusebio Di Francesco as club manager.
In extra-time, Dzeko wasted a succession of chances to score the goal that would likely have taken Roma through. He also brought them to the semi-finals of the Champions League after a memorable win over Barcelona, in which they overturned a three-goal first-leg deficit.
Roma are fifth in Serie A, three points off the Champions League berths for next season, after falling 3-0 last weekend to bitter city rivals Lazio.
The 67-year-old Italian's brief spell as manager of Premier League strugglers Fulham ended last month after only four months in charge.
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Domestically, Roma now sit fifth in the Serie A table, three points behind Inter for the final Champions League place. "It's true that there was a moment of tension, caused by the great disappointment we all experienced".
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.
Shortly afterwards, Roma forward Patrik Schick went down under a challenge from Moussa Marega in the Porto area and the referee, who consulted the video assistant but did not review the incident himself, waved play on.
Blues owner Roman Abramovich is set on Champions League qualification next season and that continues to remain in the balance.
UEFA, the European football governing body, could not immediately be reached for comment.