Eddie Jones' side clinched the Six Nations title on Saturday by trouncing Scotland 61-21 at Twickenham and now stand one victory away from a second Grand Slam in succession and surpassing the All Blacks' 18-game tier-one world-record winning run.
England must beat Ireland in Dublin next weekend to achieve those goals and Vunipola, who returned from injury as a replacement against Scotland, is eager for the team to continue improving as they seek to replace the All Blacks at the top of the rankings.
Vunipola said: "Do we want to make the next step or start again from the bottom?
"We've come a long way. I think Eddie and his coaching staff have pushed us massively to a goal that everyone laughed off at the end of the World Cup in 2015, but now it's a realistic goal.
"It all joins up. If you win 19 games on the trot you are pushing it.
"We know we're not number one yet but we want to do things within our power to show we are serious about this goal."
Should England beat Ireland at the Aviva Stadium, they will become the first side to secure back-to-back Slams since the Five Nations became the Six Nations in 2000.
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