[YVitaliy]

Код:

procedure TForm1.sbt_StatMouseMove(Sender: TObject; Shift: TShiftState; X,
  Y: Integer);
var
  pt: TPoint;
begin
  sbt_Stat.Perform( BM_SETSTATE, 1, 0 );
  with Sender as TButton do
  begin
    pt := Point(Left, Top + Height);
    pt := Parent.ClientToScreen(pt);
  end;
  PopupMenu1.Popup(pt.X, pt.Y);
  sbt_Stat.Perform( BM_SETSTATE, 0, 0 );
end;

В OnMouseMove.