The most common and most likely answer to this crossword clue is the 7 letter word CORINTH
I've seen this crossword clue in The Daily Telegraph
C
O
R
I
N
T
H
More clues leading to the result CORINTH
- My cryptic hint for old port
- Initially raking in money, half of this ancient city
- International hotel supports my city
- Greek city in the Adriatic or in Thessaly
- City that was my home before leaving there
- Bit round river? This is unseen in Greek city
- Sample decor in the Greek city
- Fancy one being given unlimited port
- My hint is not usual ancient city
- Ancient Greek city, in North, I suspect, by Cape
Buddy explains!

The straight part
We know that "Greek city" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case greek city = corinth.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.