I have an admission offer from a European Ph.D. program, and they have given me just 1 week to respond. Considering it is really early to hear anything back from the US universities (and I am keen to attend some of them over this school), how can I politely frame an email to ask for additional time, so that they don't get offended?
Just ask straight out. They won't be offended, but may not be able to hold a position open for very long. I'm pretty sure that they know already that you have several irons in the fire. If you find their offer attractive then tell them that but say that you also need to weigh other options. Perfectly normal.