According to Jefferies, 26-year-old Nish accepted in principle the club's offer of a new deal but then stalled after being advised to wait by his agent.
His contract is up in the summer, and Jefferies is eager for the Edinburgh-born forward to commit to Killie.
"I made him an offer in writing and he was happy to accept it," said Jefferies.
"It was his agent who told him not to sign. It seems like the day's gone when you're happy at a club and you accept an offer.
"You can only give a lad so long. He's got a week for that offer, then I'll have to start looking for another striker and so that offer will be taken off the table."