Pulis took charge of Wednesday last week.
“If I could take them to the Premier League, play United in a local derby at Hillsborough, that'd be my dream," said the 62-year-old.
“The success United have had – spin it round and get it for us. That is the challenge I want, it's an enormous challenge."
Pulis took the 11 managerial job of his career, aiming to bring “stability" and solve why Wednesday have plunged out of promotion contention in the last two seasons under predecessor Garry Monk.
Wednesday are currently 23rd in the Championship following a six point deduction and just six goals all season.
Pulis said: “We have an enormous following. No disrespect to Sheffield United but this club would be the biggest in Sheffield if we got into the Premier League.
“At the moment we are not. Chris (Wilder) has built United for up five years and got stability there. We need that stability.
“I'd love to place this club in a similar position, and take them to a similar position (as United) then I'd say to the chairman: Thanks very much I'm off. I'd still like United to be in the Premier League."