The decision to delay the vote until June 28 was made following objections by opposition parliamentarians, who said they had not had enough time to read the lengthy bill.
Under the draft amendment, Armenian broadcasters would have to pay the equivalent of $205 in taxes for each aired program made by a foreign media organization.
The legislation would also bar all media outlets from broadcasting on frequencies used by the state-controlled Public Television and Radio Company.
Some media observers have said that the measures are primarily aimed against RFE/RL, which broadcasts in Armenia via public and private affiliates and has a bureau in Yerevan.
In neighboring Azerbaijan, the national independent TV and radio channel, ANS, was barred from airing foreign broadcasts from the beginning of this year after authorities said it lacked the proper licenses.
ANS, previously an RFE/RL affiliate on FM, had also aired programs of the BBC and Voice of America.