The Rugby Park outfit tabled a final offer - reported to be £150,000 -having failed with an earlier attempt to sign the 24-year-old.
The fresh bid has also been rejected, with Linfield chairman Jim Kerr branding the sum on offer for the Northern Ireland international "demeaning."
The limited resources available to manager Jim Jefferies means Kilmarnock are unlikely to offer any more cash for a player who will be out of contract next May.
"Kilmarnock made an offer which was a very small increase on what they had offered previously," Kerr told the Belfast News Letter.
"Therefore the club declined to accept it.
"We have spoken at length with Peter and I can say he will be staying at Linfield for next season.
"Kilmarnock said this was their final offer, and if it is rejected, then they will wait until the end of the season to see if Peter is a free agent.
"But the offer demeaned the player and Peter is happy to stay."