Bypass Case Study by Receiving Competing Offer

Hey all, 

Been interviewing for about a month and a half now and I am currently waiting on details of an offer, it probably wont be bad but it isn't my #1 choice (we'll call it job 2). My #1 (Job 1)  on the other hand is handing out case studies soon, and we will have a week to complete it. I assume I can push off the offer expiration for a few Business days, but I doubt it would be much more than that. Do you guys have any advice on whether I could speed up the timeline of Job 1 by telling them I have an offer from Job 2 that is expiring, say early next week? 

Tldr; First Choice Job has a case study and final interview to come, Second choice is currently in offer stage, any chance to pressure job 1 to move up timeline?