And if so would this grant us any insight into the relations between P, NP, and EXPTIME?
We already know that NP is included in EXPTIME. So every NP problem can not only be reduced to an EXPTIME problem: it is an EXPTIME problem!
Any problem having an EXPTIME algorithm can be reduced to an EXPTIME problem, namely the problem itself (the reduction maps the input to itself). Every problem in NP has an EXPTIME algorithm, so every problem in NP can be reduced to some EXPTIME problem in polynomial time.