Late on Friday, the White House issued a statement saying Trump would sign the bill after reviewing the final version, Reuters reported.
The US Senate had voted almost unanimously on Thursday to slap new sanctions on the three countries.
By signing the bill into law, Trump cannot ease the sanctions against Moscow, Tehran and Pyongyang unless he seeks congressional approval.
On Friday, the US Treasury Department announced new sanctions on Iran over what it called Tehran's "continued provocative actions," including Thursday's successful test of a satellite-launch rocket.
The legislation was also in part a response to conclusions by US intelligence agencies that Russia allegedly meddled in the 2016 US presidential election.
Russia had been threatening retaliation for weeks. Its response suggests it has set aside initial hopes of better ties with Washington under Trump, something the US leader, before he was elected, had said he wanted to achieve.