Jared Kushner has testified before the Senate investigators saying “all of his actions were proper”.
Trump’s willingness to support the measure is an acknowledgement that he has yet to sell his party on his hopes for forging a warmer relationship with Moscow.
In a statement late on Friday, White House spokeswoman Sarah Huckabee Sanders said Trump had “reviewed the final version and, based on its responsiveness to his negotiations, approves the bill and intends to sign it”.
Trump’s decision followed negotiations on “critical elements” of the bill, Sanders said, without specifying what those elements were.
The bill passed Congress with overwhelming support on Thursday, dashing Trump’s stated hopes for warmer ties with Russia.
It bars Trump from easing sanctions against Russia unless he seeks congressional approval.
Trump had privately expressed frustration over Congress’ ability to limit or override the power of the president on national security matters, the Associated Press news agency reported, citing anonymous Trump administration officials.
But faced with strong bipartisan support in the House and Senate, the president had little choice but to sign the bill into law.
Still, approving a bill that penalises Russia’s alleged election interference marks a significant shift for Trump.
He has repeatedly cast doubt on claims that Russia sought to tip the election in his favour.
The 184-page bill seeks to hit Russian President Vladimir Putin and the oligarchs close to him by targeting Russian corruption, human rights abusers, and crucial sectors of the Russian economy, including weapons sales and energy exports.
It also imposes financial sanctions against Iran and North Korea.
Meanwhile, some European countries expressed concerns that the measures targeting Russia’s energy sector would harm its businesses involved in piping Russian natural gas.
Germany’s foreign minister said his country would not accept the US sanctions against Russia being applied to European companies.
A European Commission spokesman in Brussels said the bloc would be following the sanctions process closely.