German European Affairs minister Michael Roth said the stimulus is crucial for many European countries whose economies have been devastated by the pandemic.
Both Poland and Hungary, which have conservative, nationalist governments, have said they fear the EU mechanism will be used to punish their values.
Many projects for Poland and Hungary — which are already being formally investigated by the EU for their potential violations of the rule of law — could be held up.
Under the enhanced cooperation procedure, a group of EU nations can decide to move forward in situations where all 27 countries are not on the same page.
If the standoff continues, such a move could at least help unlock the bloc's 750 billion-euro ($909 billion) economic recovery package.