This is known as an exploding offer.
So what should you do if you receive an exploding offer?
Exploding offers are anathema to your ability to effectively navigate the labor market. Thus, there is only one thing to do. Treat the offer as a non-offer unless the expiration window is widened.
In no uncertain terms, convey that if the offer is exploding, it’s useless to you.
Example conversation:
I have one big concern. You mentioned that this offer explodes in 48 hours. I’m afraid this doesn’t work at all for me. There’s no way that I can make a decision on this offer within a 48 hour window. I’m currently wrapping up my interview process at a few other companies, which is likely to take me another week or so. So I’m going to need more time to make an informed decision.
If they push back and say this is the best they can do, then politely reply:
That’s really unfortunate. I like [YOUR COMPANY] and was really excited about the team, but like I said, there’s no way I can consider this offer. 48 hours just too unreasonable of a window. The next company I join will be a big life decision for me, and I take my commitments very seriously. I also need to consult with my [EXTERNAL_DECISION_MAKER]. There’s no way that I can make a decision I’m comfortable with in this short an amount of time.
Pretty much any company will relent at this point. If they persist, don’t be afraid to walk away over it. (They probably won’t let that happen, and will come grab you as you’re walking out the door. But if they don’t, then honestly, screw ‘em.)
I was given several exploding offers during my job search. And every time, I did essentially this. Every single offer immediately widened to become more reasonable, sometimes by several weeks.
I want to emphasize, lest I be misunderstood here—what I’m saying is not to just silently let an exploding offer expire, and assume that everything will be fine and they’ll still hire you. They won’t. For exploding offers to be a credible weapon, a company has to have a reputation of enforcing them. I’m saying explicitly call this out as an issue when they make the offer.
Don’t let a company bully you into giving away your negotiating power.
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https://haseebq.com/my-ten-rules-for-negotiating-a-job-offer/
But this is even far worse than an exploding offer.
Never accept an offer by itself. A verbal offer is worth nothing. The same goes for a written offer. What he needs is an actual written contract with things like actual start date, salary details, relocation package, details of benefits, etc.
If he makes a one-way commitment without making them commit to anything in return, he removes the urgency to lock him in as a candidate, he makes himself look desperate, he incentivizes them to delay the final contract for as long as they can, and he frees them to explore other potential candidates in the meantime. At the same time, by signing such a promise, that neutralizes his ability to negotiate, so it allows them to potentially reduce the salary and reduce the benefits they were originally willing to give him. This is in addition to the exploding offer nonsense, which reduces his ability to receive competing offers. In short, it's an all-around very bad idea for the candidate.
And one last thing, you said a "headhunter"? Do you mean to say that the person is a 3rd party recruiter? If he's negotiating with a 3rd party recruiter, he's negotiating with the wrong party. He should contact the company directly. By negotiating through a 3rd party, any promise/concession made by the headhunter can easily be reneged on because the employer can just claim that the headhunter was not authorized to make such a promise/concession in the first place. But of course, the reverse won't be true, you can be assured that any promises/concessions the candidate makes to that 3rd party will be remembered and noted down and could potentially be held against him by the new employer. So it's not only a waste of time negotiating with a 3rd party headhunter, but it's highly counter-productive to do so as well.