Captain Mark Byrne and top scorer Scott Boden are two of six players who have been offered fresh terms but have yet to put pen to paper.
"There are deadlines there," said Feeney. "If you want to play for this club then you want to sign what's been offered, you take it.
"If you don't then you move on."
The futures of Alex Rodman, Danny Holmes, John Christophe-Ayina and Medy Elito are also in doubt.
So far just defender Darren Jones has committed to a new one year contract while Andrew Hughes rejected his offer to sign for League One side Peterborough.
Football League rules state that players must notify clubs in writing within a month whether they intend to accept their terms, with County having released their retained list on May 10.
Midfielder Byrne has been linked with a reunion with former Newport manager Justin Edinburgh at Gillingham after similar speculation during the January transfer window.
Feeney accepts the longer the situation is drawn out, the more likely it is that Byrne and others could leave when their current contracts expire at the end of the month.
"That could be the case," Feeney added.
"At the end of the day I can't sit and wait, I'm talking to people.
"We'll see where there are but if they wanted to sign contracts, get it over the line and get it done.
"They'll wait so I'm in a position where I can wait.
"I'm talking to people and the deadline is coming up so we're going to have to possibly move on."
Feeney is confident of securing a further three new signings in the coming weeks.
Midfielder Joss Labadie became their fourth addition this summer joining Scot Bennett, Mark Randall and Jamie Turley in signing two year contracts to move to Rodney Parade.
Manager Warren Feeney has told Newport County's out of contract players to "sign what's been offered" if they want to stay at the club.