An Offer Is In, and the Clock Starts Ticking

A few months ago, I subbed a PB manuscript to a half dozen publishers. An assistant editor at one of the publishers contacted me to let me know they may be interested (she's passing it up to the to the head honcho). It's a very small, but reputable, house. With some luck, I'll get an offer from them. I imagine that if I had an agent, at that point he or she would contact the other publishers where the manuscript was subbed to give them a chance to offer or pass. Would it be ok for me, an unagented slush pile warrior, to do it for myself?

Yes. Get in touch with them to let them know you've received an offer from X house, and you'll need a response by X date (within a week of receiving the first offer). That's all you need to say, but if you want to grease the wheels with some flattery ("I'd still love to work with your house, etc"), that's also among the things agents do.