The agreement would be aimed at protecting Russia's "national interests," the TASS state news agency reported, citing the document, on Wednesday.
A foreigner entering Russia would be prohibited from "interfering with the activities of public authorities of the Russian Federation, discrediting in any form the foreign and domestic state policy of the Russian Federation, public authorities and their officials."
Reuters could not independently verify the draft bill. The Internal Affairs ministry did not immediately respond to requests for a comment.
Since its full-scale invasion on Ukraine in February 2022, Russia has imposed a number of restrictions on foreigners from what it calls "unfriendly countries" - meaning those that have imposed sanctions on it over its war in Ukraine.
The internal affairs draft bill provides that foreigners would be prohibited from disparaging or inciting the denial of "significant moral" values, such as marriage as a union of a man and a woman, family, as well as from disseminating propaganda of non-traditional sexual relationships.
Distortion of the "historical truth" about the Soviet people's defence of the country and its contribution to the victory over fascist Germany in World War Two would also be prohibited, TASS reported.
TASS did not specify what repercussions foreigners would face if they broke the agreement.
For the draft to become law, the document has to be introduced to the State Duma, Russia's lower house of parliament, and go through committee review and several readings before being submitted to President Vladimir Putin for signing.