Depends on how much they want you, they might accommodate. Obviously this is unlikely to work if you're fresh out of school and have no track record to speak of.
You are not asking for anything extra here, you're just asking, de facto, for conditions that would let you value the options part of your compensation at anything greater than zero. You got dinged for it on the base salary after all, and it costs the company very close to zero to give you this paper wealth. If the contract is abusive, you can be sure you'll be treated 100% in line with the letter of the contract and your options are very likely to be worth nothing.
Don't be afraid to offend, this is just business. Don't speak in confrontational tone either though, and be prepared to walk away, just like in any other negotiation. The worst outcome is nothing changes, but you had that to begin with. If you're aggressive, the second move will be an email from the company lawyer telling you the changes you've suggested are out of the question. After that, if you're prepared to walk away, they will agree to a part or all of it.
The net benefit of this is you could introduce numerous improvements into your option contract if you succeed, i.e. eliminate or weaken the clawback clauses, negotiate early exercise, take care of accelerated vesting or option replacement in case of an acquisition (otherwise you could be left with literally nothing after the exit), vastly extend the exercise window (from the default 0-90 days after you leave to many years), negotiate that you get to keep shares if you're _fired_, etc, etc. A good lawyer could write a thick tome on this, and you're NOT going to be able to learn it in your spare time.
If equity is substantial, and you think it might eventually be worth anything, I also advise hiring a tax accountant with experience in such matters. There are different types of stock options, with different levels of taxation (conditioned on your exercise strategy). Under some strategies you could end up on the hook for six-seven figure amounts of tax even though you can't sell your shares, and might never be able to do so.
But for most startups, given the low probability of success and the many ways you can be screwed (dilution, clawback, liquidation preferences, etc) you'd need to have A LOT of faith in your startup (and in the soundness of your options contract) to exercise early.
If you do nothing else, try to negotiate that: a). You're not required to sell _vested_ stock back to the company if you leave / or even get fired, b). You get _something_, preferably full, immediate vesting, in the event of an acquisition, c). Your exercise window spans many years after you leave, so that if the startup does exit you can get something for the vested portion of your option without exercising it before you leave (and potentially exposing yourself to massive reaming by the IRS).
None of the above is a replacement for proper legal / tax advice though, it's merely a distillation of the advice I received that was applicable to my particular circumstances at the time.