The sources said differences in opinion between board members from parent company Hon Hai and those from the original Chimei before its merger with Hon Hai unit Innolux last March were the cause of the lack of agreement.
The precise area of disagreement at last week's meeting was not clear, but one of the sources said it could be over the form of the relationship between the companies after the spin-off.
The board will meet again on the plan at an unpecified date.
The company said last month it would make the two units into separate companies. The move is aimed at gleaning bigger gains from businesses expected to be one of its fastest growth engines.
Other sources had told Reuters in January that the initial public offering of the touch panel unit would be in the second half of this year.
Touch panels are one of the hottest areas in technology at present thanks to the rapid growth of gadgets like the iPad and iPhone. Hon Hai is one of the biggest manufacturers of the Apple products.