Well we have an agreement to meet people the other are dating before they become sexual partners. First meetings of metamours have included a 2 minute greet at the door when I date picked one of us up, a board game night for my husband and I with his gf and her husband, meeting one of his girlfriends for drinks just her and I, having lunch with my husband and his date, meeting my bf's wife for coffee just her and I. It sounds like you are not comfortable with that idea yet, but that's just an example of one way that stuff can be managed.
I don't know why you think only women would want to check that out - isn't it a good thing he wants to make sure that your husband is OK with things? I think that's nice and SMART especially as you are not experienced at seeing other people but new, and new can come with a boatload of issues. I'd give him your husband's email and let him initiate contact if your husband is too awkward with the idea of phone or meeting in person (nothing is wrong with sticking to email if that's his comfort level). Truthfully I would not date somebody who's live in partner was too uncomfortable to at least exchange a brief email or two with me to acknowledge everything is on the up and up and dating me isn't a problem.
Happiness will never come to those who fail to appreciate what they already have.