The highly-rated midfielder, 18, has risen through the ranks at his hometown club and duly enjoyed his first-team breakthrough this season, appearing in seven of the Essex outfit's eight matches to date.
With Gilbey's contract expiring next summer Colchester had hoped to tie him down to new terms, but it appears that will have to wait for the meantime at least.
"We've offered Alex a deal and his representatives have turned it down," manager Joe Dunne told the Daily Gazette.
"It's not a concern - they have to do what's best for them and we as a club have to do what's best for us.
"We've offered what we think is right, in terms of our structure and we think it's fair. But we also appreciate that in the modern climate, you have to do what's best for yourself.
"It's certainly not something that has affected Alex or his football, nor our relationship.
"We'll wait and see what happens but Alex is someone who I've known for a long time now and I gave him his debut."
- Sports & Recreation