The two countries signed a cooperation agreement in December last year to expand the Paks nuclear plant in Hungary.
Under the agreement, Russia will provide a loan of $11bn to Hungary.
According to the government, the latest legislation is in line with similar laws in other European countries.
The Associated Press has quoted the Hungarian Government as saying that the contract is a ‘marriage of interests’, and the financial terms offered by Russia could not be matched by any other supplier.
However, the legislation is being opposed by political parties and civic groups, who contend that the law is an attempt to conceal a corrupt deal.
European Commission spokeswoman Lucia Caudet told the Wall Street Journal that awarding the Paks nuclear power plant expansion project without a tender procedure has been brought to the attention of the Commission.
"Commission services are currently examining this issue and clarifying it with the Hungarian authorities," she added.
The €12.5bn ($14bn) expansion project will have two reactors of 1,200MW each. Construction is expected to commence in 2018.
The first reactor is expected to go online in 2025 and the second a year later.
Paks plant currently meets 50% of the country’s electricity demand.
