Lewis Emanuel has been handed a "take it or leave it" offer by City.
A new one-year contract is on the table for the young defender and the club have made it clear that there is no room for negotiation.
City would like to hold on to the home-grown Emanuel, who can operate either in left midfield or at full back. But the onus is on the 22-year-old and it is doubtful whether he will consider the offer enough.
Chairman Julian Rhodes said: "Maybe there are other clubs who can pay more money than we can. It would be a shame but that's the way it is I'm afraid.
"Lewis has had a good end to the season and we would like to keep him. But we cannot go down the route of offering more than we can afford.
"I've spoken to the agent and said what we could do and now it's up to Lewis."
Colin Todd is moving closer to unveiling his first signing of the summer.
An announcement about the former Premiership midfielder is expected within the next couple of weeks.
But that is likely to be the end of the transfer activity until Rhodes and Peter Etherington have signed their power-sharing deal.
Rhodes added: "Terms have been agreed with the player and it will probably happen before the end of the month.
"After that, we're waiting for Peter and I to get things sorted out.
"The budget for Colin has been set low to start with but that will change if we get our deal done.
"He's been given two different figures depending on that. There is not a huge difference between them but it is noticeable.
"Colin is looking at some interesting signings at the moment but it's likely that there's nothing we can do until the end of June or early July.
"Ideally, you'd get them all in for July 1 and away you go but that probably won't be possible.
"It's not the end of the world. Since I've been involved with the club, the season we've made the latest signings was 1998-99 when we didn't get the finance in place until the end of July.
"I remember signing the likes of Lee Mills the day before the season started and it took time to settle. We took only five points from the first seven games but so what, we got promoted at the end of the season."
l City had the sixth-highest crowds in League One last season.
The Valley Parade figure of 8,264, despite being 600 down on the previous year, was still well above the division's average. Nottingham Forest were the best with a staggering 20,257 while promoted Colchester managed only 3,969.
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