Someone sent me a message saying that you are also my suitor. Is he chasing me or asking me to pursue her?

First of all, the "she" and "he" in your question are different genders, so they are a little confused.

In addition, "you are also my pursuer" should generally be understood as that he or she thinks you are pursuing him or her.

Finally, the "ye" in this sentence shows that this sentence has a premise: you are just one of the pursuers.