Woolford, 25, has been a key member of the Iron team since moving from the non-league Minstermen in 2008.
Speculation increased on Wednesday that Nottingham Forest were interested in the player, with the Daily Mail claiming a £400,000 bid had been tabled for him.
Ian Baraclough, the Iron manager, said there had been no contact with the Midlands club.
He told BBC Radio Humberside that a fresh contract had been offered to the player, who is out of contract in the summer, but "he doesn't want to discuss his future at the moment".