Very likely that the report is either (a) completely false, or (b), true, but the Patriots are just trying to drive up the price, as they see contenders as bigger threats than the Jets.
In the very unlikely scenario that this is true, there is only one reason why this could happen. Belichick obviously knows the Jets would want to avoid trading their best player to their most hated rival, so if there truly is interest, the Patriots must have something the Jets want. And that would not be draft picks, etc, since any team could offer those types of assets.
Ryan Mallett. The Jets would want a franchise quarterback for a low price, and if Idzik believes that Mallett is a franchise quarterback, then the Patriots could offer the Jets something that other teams cannot. Getting a quarterback would also balance the potential damage of the Patriots getting Revis.
When you look at this year's quarterback class, this could be an outside possibility.
Let me emphasize that I think there's a 90% chance that the report is BS to begin with, but it's fun to speculate.