Generally a counteroffer accepts some terms of the previous offer while proposing changes in other terms. Each counteroffer can lead to another, but the party that proposes a counteroffer has rejected the previous offer which is legally voided even if it would have been binding prior to the counteroffer.
Make some kind of reasonable counteroffer:’ This was the tricky part.
Scott Pratt -- An Innocent Client
Thank you for the generous counteroffer, but we must decline.
Henry H. Neff -- The Second Siege
To a counteroffer.
Robert Ludlum -- The Bourne Ultimatum
I know that you asked me to tell you because you wanted to have a chance to make me a counter-offer.
Ayn Rand -- Atlas Shrugged
She refused to pay the price named, and after a moment’s hesitation, met it by a counter-offer of half the amount.
Edith Wharton -- The House of Mirth
The pair went back and forth in rapid succession, making offers and counteroffers until Ms. Richter finally agreed to let David have three Promethean Scholars along with four spiritwracks of his choice.