Manager Paul Ince was sacked as manager on Tuesday following a poor run of form in the Championship, casting greater uncertainty over the future of his son.
Tom Ince will be out of contract in the summer and has been linked with a move away from Bloomfield Road for some time, with a number of sides thought to be interested in securing his services.
Swansea City, Everton and Crystal Palace are among those monitoring the situation, while speculation has also suggested that Monaco are keen to strike a deal.
Oyston says there have been no bids yet for the 21-year-old and he is not sure if anyone will make contact before the end of January.
If there is to be no transfer this month, the club will have to wait until the summer to see if the player wants to stay or go.
"There are three options, all of which we are okay with," Oyston told Sky Sports News.
"We were very open and up front with Tom about that, and vice-versa I have to say.
"That is a decision for other clubs. If anyone wants to take any of our players, then they will be in touch, but we have had no formal contact yet."