A "counter offer" is when someone responds to an offer or proposal with a different offer of their own. It's like a negotiation where both parties propose different terms until they reach an agreement. Full definition
You don't want two sales associates making counter offers. (petbusiness.com)
We can give you samples of counter offer letter templates should you be needing samples in creating this kind of document. (template.net)
This approach allows the employer to quickly wrap up negotiations without a number of counter offers going back and forth with the prospective employee. (thebalance.com)