Shaw was new manager Louis van Gaal's second signing of the close-season, joining United from Southampton in a big-money deal at the end of the World Cup.
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However, the 19-year-old recruit is taking everything in his stride, having already amassed 60 senior appearances for Southampton.
"I don't feel any pressure to come in for Evra," Shaw told reporters ahead of United's opening International Champions Cup match against LA Galaxy on Wednesday.
"He has been great for the club for the last 10 years and I hope I can carry that on.
"There is always going to be pressure, no matter where you go, but I try not to focus on that. I try to focus on my own game, what I do on the pitch.
"And with the new gaffer you have to impress on the pitch and that is something I am looking forward to - hopefully in these pre-season games I am looking to show him what I can do."