The unbeaten American retained his world light-heavyweight titles on Saturday after a stoppage victory over Sergey Kovalev in their rematch in Las Vegas.
Ward first stepped up from the super-middleweight ranks to face the Russian last November and the 33-year-old has now admitted a bout against the WBA and IBF heavyweight champion could tempt him to make the step up to the top division.
"I always seem to be knocking down giants one by one," Ward said. "Heavyweight, that's always been a dream of mine. It's got to be against the right guy, but I dream big and anything is possible."
Ward added: "He [Ward's trainer Virgil Hunter] has always been setting the bar high, that is the essence of our team right there.
"That man will have me believing things that I never even thought of. Like, 'Anthony Joshua, really?' and it will make perfect sense to him. So we'll see."
Hunter first suggested the idea of Ward facing Joshua and believes his man has the attributes to outbox the British knockout specialist.
Ward would also not be the first fighter to make the leap, with the legendary Roy Jones Jr having moved up from the light-heavyweight division in 2003 to defeat John Ruiz for the WBA heavyweight title.
"Fighting a bigger man doesn't have to do with size, it has to do with your attributes and their attributes," Hunter said.
"If you are able to single out one thing that makes it a competitive fight for yourself, you take that chance.
"I'd like to see him fight Anthony Joshua. I'm not playing, I'm serious. I think it would be a very interesting fight. I believe he could outbox Anthony Joshua.
"There's some things about Joshua that I see. He's a good kid, I know him, but I see things with him in particular and he is the best [of the heavyweight champions], but styles make fights.
"It would be a waste of time to go to the cruiserweights, we should just leapfrog it, go for it all."
