The 24-year-old hopes the Saints can overcome Sunderland on Saturday to reach a semi-final at Wembley, having already played a key role in their cup run
Mauricio Pochettino's side face Sunderland on Saturday in the fifth round of the competition, with a Wembley final only three games away.
Cork has helped Southampton to FA Cup victories over Burnley and Yeovil Town on their way to the last 16 and he wants to extend their eight-game unbeaten run in all competitions this weekend
"It would be nice and I would love to get as far as we can," he told the club's official website.
"The semi-final is at Wembley so right now anywhere near that sort of stage would be brilliant and I know that the lads would love it too.
"If we do play as well as we have done in the league in the cup games then we do have a chance.
"This will hopefully be a chance for us to get into the quarter-finals and then it's only one win from Wembley and then from there the final.
"It's exciting just thinking about it but I do that every year when we beat two teams and then go out, so it would be nice to go as far as possible this time."