tooltip-frame-parameters

tooltip-frame-parameters is a variable defined in `tooltip.el'.
Its value is

((name . "tooltip")
 (internal-border-width . 2)
 (border-width . 1))

Documentation:
Frame parameters used for tooltips.

If `left' or `top' parameters are included, they specify the absolute
position to pop up the tooltip.

Note that font and color parameters are ignored, and the attributes
of the `tooltip' face are used instead.

You can customize this variable.