the article will be something like: TJ is a RB over 30yrs old.... this means the Pats are interested.
After 4 games this season half of this board never wanted to see Jones on the field again. After the Indy game he was unilaterally trashed even though he was clearly playing hurt (and the secondary cost them that game anyway.) Lots of people seem to dislike Jones on a deeper level than thinking that he needs a paycut.
That may be why this rumor surfaced. Belli is thinking - Let the Jets think if they cut him he will head to their rival and turn into a bad PR move. That way the Jets may hold onto to a 30 year old RB
I think what actually happened is that Tannenbaum knew Bellichick would try and start this rumor exactly for the reasons you stated, so he started them himself to make BB go nuts figuring out who the traitor within their organization is. BB will be losing precious hours he could have been spending on evaluating potential draft picks. Brilliant move by Tannenbaum.