I agree with the idea that there may be a lot of issues brewing in your marriage and possibly carry over from past relationships that could be part of this. Is your wife ready to go to counseling before jumping into this relationship she gave you an ultimatum about? And what if you said its him or me? Would she still want to just have that relationship or does she value you enough to slow the whole thing down?
In my marriage, a big issue was infertility and all that carries, and the infertility was and (still is ) despite birth of our wonderchild who I am currently home with, and with help from his grandma. Also the infertility is due to meds I take for bipolar disorder and it is really questionable to my mind that I need to have been on them and be on them without a break for as long as I have (and am).
The issues combined to get my husband interested in a swinger's board, because he says he feels emotionally fully attached to me, but wants more of a sex life than between above issues noted I have been able to give him or myself. The way I am wired, I actually would feel more comfortable for him to have a long term girlfriend that I am also friends with (sexual for him but not sexual for me). And although he has reluctantly said that might work out, lets put it this way, neither of us is in a rush to go out and find anybody. Also, if my body's ability to have a sex life suddenly was returned to me, I would reserve my right to be able to do what he is allowed to do. My husband has agreed that our rights are equal in this, whatever we end up deciding to do.
He also told me, if I want neither of us to see anyone he is ok with that too.
So your wife really has to 1) respect your boundaries, 2) you respect hers and 3) you need to see a marriage counselor. (which i and my husband did).