The Greece international, 34, is a free agent after being released by Bolton in the summer and attended Hull's 2-2 draw with Everton in the Barclays Premier League.
Tigers boss Phil Brown said: "We are currently talking to Stelios with regard to a contract until the end of the season or possibly until the January transfer window.
Hull City boss Phil Brown is in talks with former Bolton midfielder Stelios Giannakopoulos
"Stelios has got an ambition to play for Greece in the World Cup but is currently without a club.
"As far as I am concerned if we can convince Stelios to sign for us it would be a no-lose situation."
Signing Stelios would bring to 13 the number of new players brought in by Brown since the club secured promotion to the top flight last May.
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