The Hammers could qualify for the Champions League for the first time ever this season.
Slaven Bilic's team are currently a point off the Champions League places in fifth, and are a victory away from an FA Cup semi-final appearance at Wembley.
Captain Mark Noble admitted to TalkSPORT last week that he would rather bring home the trophy than make it into Europe's premier club competition next season - adding that Sullivan and fellow co-chairman David Gold would "probably kill me" for that comment, with a top-four finish bringing far greater financial rewards.
And Sullivan admits he would choose the Champions League over football's oldest cup competition, but only on one condition.
"If you said third or the FA Cup I’d go for third because that’s automatic Champions League qualification," the 67-year-old told TalkSPORT's Alan Brazil Sports Breakfast show on Monday. "I think third’s virtually impossible - Arsenal had faltered but the win against Everton the other week was a huge one.
"Fourth only gets you a tough qualifying match so I would take either and be absolutely elated. To be honest, if the season finished now it would be a fantastic season for us."
Finishing fourth would see West Ham face the possibility of playing the equivalently-placed team from Spain or Germany (currently Villarreal and Schalke 04) or the third-best team in Italy or France (Roma and Nice) in a two-legged tie, with the winner progressing to the Champions League group stages.
The Hammers have never before qualified for the competition in any format.