The most common and most likely answer to this crossword clue is the 7 letter word HARVARD
I've seen this crossword clue in The Sunday Telegraph
H
A
R
V
A
R
D
More clues leading to the result HARVARD
- Difficult to get hold of a particular Bible for US university
- University serious about a bible
- Very able recruits initially residing in tough US university
- Difficult to swallow odd bits of Viagra in this university
- Virginia right to get a place in tough university
- University a challenge, Aussie PM virtually admitted
- Difficult securing first in Richmonduniversity
- Difficult to comprehend airman recruit against the N American T-6
- Firm about a sort of Bible university in America
- Where some top Americans learn it’s difficult bringing in a lot of change
Buddy explains!

The straight part
We know that "university in America" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case university in america = harvard.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.