A-3 B-4 and vice versa
A-7 B-8 and vice versa

A-2 and B-3
A-9 and B-8

Then only A able to find the number.

The four solutions are:
A:2, B:3 ;
A:3, B:4 ;
A:9, B:8 ;
A:8,B:7

EXPLANATION-
Consider the numbers between 1-10:     1, 2,3,4,5,6,7,8,9, 10

The fact that neither of them knows the other person’s number so they cannot have the extreme values, that are, 1 & 10.
So we are left with 2-9.

If the numbers known to B were 2(or 9), he could have immediately deduced the fact that the number known to A is 3(or 8). But since, he says “Nor do I know your number”, it means that number known to B is not 2, 9.  HOWEVER, this does not mean that the number known to A can’t be 2 or 3.

So, A’s number can be one of : 2, 3, 4, 5, 6, 7, 8, 9
And, B’s number can be one of : 3, 4, 5, 6, 7, 8

Now, if A’s number is 2, then he can be sure that B’s number is 3,
if A’s number is 3, then he can be sure that B’s number is 4,
if A’s number is 9, then he can be sure that B’s number is 8,
if A’s number is 8, then he can be sure that B’s number is 7.