Shaw was new manager Louis van Gaal's second signing of the close-season, joining United from fellow Premier League club Southampton in a big-money deal at the end of the FIFA World Cup.
The England international is expected to shoulder the load at left-back next season after Evra - winner of 15 trophies in eight years at Old Trafford - completed his move to Serie A champions Juventus, despite signing a new deal in May.
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 said ahead of United's opening International Champions Cup match against the Los Angeles 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, and hopefully in these pre-season games I am looking to show him what I can do."
