Negotiating Job Offer with a Counter Offer

I work in finance. I've recently received a job offer for $100k + 20% bonus. This is a significant bump to my current pay (~30%) and I was happy to take it. I brought it up with my current employer and they counter offered me $115k + discretionary bonus (~12-20%). I have little intention to stay with my current employer but would like to know how best to bring this up with the other company.

I haven't yet signed their offer letter. I am thinking of requesting $115k + 20% or $111k + 25%, their choice, and let them know I have received a similar offer. I am not sure whether to mention this is a counter offer from my current employer, but it seems odd not to mention it since I said I wasn't interviewing elsewhere when in discussions with them. I am also unsure about providing two alternate counter offers to them - but I like the idea.

The jobs are different, with my current job known to be better paying. I also feel my target company is at the top of their range. Is it unwise to push this hard? Or should I just let them tell me they can't do it?