United will be able to utilise the top level experience of ex-Juventus playmaker Katie Zelem, Alex Greenwood and Siobhan Chamberlain this season, but former England defender Stoney is keen to keep a lid on expectations.
They open their second tier Championship campaign against Aston Villa on Sept. 9.
"We would be naive to think we can go and win everything," Stoney told ESPN.
"We're literally six weeks in. We had a good start on Sunday but it's game by game. It's a cliche, but it is. We're learning as we go and we're going to make mistakes, we know that.
"There's no pressure from the club to go up in year one (promotion to the FA Women's Super League) but if you ask the players, they will give you a different answer."
Zelem, who won the league with Juventus last year, said a historically successful club such as United will always be under pressure to deliver trophies.
"It's Manchester United and we know the expectation that comes with that," the 22-year-old said. "We want to win the league, get promoted and get into the Champions League."
United next face Reading in the League Cup on Saturday.
(Reporting by Hardik Vyas in Bengaluru; Editing by Toby Davis)
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