If adopted, it would force NGOs that receive more than 23,000 euros ($24,000) annually from a foreign source to register with a court as a "foreign-supported organisation".
NGOs would also have to label themselves as a "foreign-supported organisation" on their websites and publications.
The law, first mooted in January when an official in Prime Minister Viktor Orban's ruling Fidesz party said foreign-funded NGOs should be "swept out" of Hungary, had been delayed after widespread criticism, including by the European Commission and United Nations experts who urged its withdrawal.
The June 13 date was finally confirmed on the parliament website Thursday.
Last week the EU's rights watchdog Venice Commission said that while the draft bill "pursues legitimate aims" it is "excessive" and called on Budapest to consult with local NGOs affected by the legislation before the vote.
It also accused "some state authorities" of staging a "virulent" campaign against foreign-funded NGOs.
Budapest said Thursday that it is taking the Venice Commission concerns into account and will amend parts of the draft bill, for example dropping the requirement for all foreign donors to be named on a group's publications.
The amendments were dismissed as "mere fine tuning" by the co-head of a local NGO often criticised by the government, the Hungarian Helsinki Committee refugee rights group.
"As before, NGOs could still be closed down if they fail to comply with the new rules, while no consultations will take place before the vote," Marta Pardavi told AFP.
"The general intent to stigmatise also remains," she said.
Since his reelection as Hungarian premier in 2014 Orban has regularly accused foreign-funded NGOs, in particular those receiving support from US billionaire George Soros, of interfering in Hungarian politics.
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