The law under consideration in Moscow would ban the dissemination of information among minors that promotes homosexuality.
Parliament's lower house intends to pass the law by the end of the current session, RIA Novosti reported.
"The EU called on Russia to refrain from adopting federal legislation on 'homosexual propaganda,' which it believed could increase discrimination and violence against LGBTI individuals," the EU said in a statement on its website.
The EU also said it will monitor Russia's treatment of non-governmental organizations in the country.
Russia passed a law in November requiring NGOs involved in political activity to register as foreign agents. Russian officials said the law is intended to protect against foreign interference in Russian politics.
NGOs have complained labeling them foreign agents allows the government to spy on them, and some have refused to register.
The law has since been used by prosecutors to open cases against NGOs engaged in activities such as health projects and wildlife conservation, RIA Novosti said.