I’m an advocate of transparency, but you must use extreme caution that you don’t look like you’re trying to start a bidding war.
By telling a potential employer that you’ve already received another offer shows them you’re desired and may not be available on the job market much longer. However, you want to time it right. I recommend that you wait until the end of the final interviews. By that point, you may know where you stack up or have an idea if they will extend an offer.
I recommend you say something like, “I’m so excited about the possibility of working for your company. Thank you so much for your time and for considering me this far. I think I will be able to make an immediate impact and add value. I wanted to make you aware of something that developed this week and see if you have a recommendation for me. I’ve received another job offer. While you are by far my top choice, there are aspects of the other role that interest me. They’ve asked me to give them an answer within the next couple of days. Do you anticipate that you’ll be making a decision soon?”
At the very least, you’ve been honest. If they want you, they will accelerate their decision, so they don’t lose you to the other opportunity.