To my knowledge, both of them imply that someone thinks of something in a short time, but is there any difference between them? For example
It never crosses my mind that he is lying to me.
It never crosses my mind that we might lose.
I think using enter my mind here is fine.
It never enters my mind that he is lying to me.
It never enters my mind that we might lose.
Best Answer
The two verbs 'to cross' an 'to enter' are the key here:
(Free Dictionary)
The connotations of cross are to go in and then out of, in your case, one's mind. Compare this to enter:
(Free Dictionary)
Enter simply denotes the going in to one's mind and has no such connotation of it being fleeting.
Overall, the difference in verb choice makes crossing one's mind into just a brief thought - it comes and then goes - whethers enter, just means that a thought has gone into your mind but not out again.