We need to do nothing. Like teeth, ignore them and they'll go away. If they want him, they'll bid accordingly. If they don't, they'lll bis accordingly.
What makes or breaks this is Rodgers.
If he shows faith in Carroll, at least publicly, and gives him starts in all three cups, plus a decent go in the league as a Plan B, he'll show Newcastle's bid for what it is and tell Carroll, they can't want him much, otherwise they'd bid a figure which would allow a sale to go through.
Also, remind him that they sold him against his will and that 18 months ago they rated him at £35M, now they think he's worth only a third of that. In other words, use Newcastle's figures against them, to show that they didn't and don't really want him badly enough.
Point out, Allen. A player who had only one year of PL experience, but we believed in him and bid accordingly.