Russian officials have expressed "concern" over a US anti-doping bill that grants US officials the power to prosecute individuals responsible for doping conspiracies.

The legislation - called the Rodchenkov Anti-Doping Act - received formal approval in the US Senate on Monday.

The bill "evokes nothing but concern", Kremlin spokesman Dmitry Peskov told reporters on Tuesday.

"We are highly critical of any attempts by the United State to spread its jurisdiction to other countries," Peskov added.

The Rodchenkov Act is named after whistleblower Grigory Rodchenkov, the former head of Russia's national anti-doping laboratory, who set in motion the Russia doping scandal in 2016.

Rodchenkov was heavily involved in a state-backed conspiracy designed to cover up Russia's cheating at the 2014 Sochi Winter Olympics and now lives in hiding in the United States.

The bill now requires a signature from the US president before it becomes law.

Unlike the World Anti-Doping Agency (WADA) which can sanction athletes, the "Rodchenkov" Act targets the entourage of athletes - coaches, agents, managers -- and can lead to fines of up to $1 million or a prison term of up to 10 years.

A WADA spokesman on Monday expressed concern that similar bills could emerge in other countries for political reasons, for example targeting athletes from specific nations in retaliation. 

The global anti-doping watchdog also suggested that the law may deter whistleblowers from coming forward if there is risk they could be persecuted. 

United States Anti-Doping Agency (USADA) chief executive Travis Tygart has insisted that the legislation offers greater protection for both clean athletes and whistleblowers.