Lawmakers said the executive order, which would give the president decision-making power on imposing sanctions, was insufficient.
Mr Bolton stated the order is meant to "protect the United States from foreign interference in our elections and really our political process more broadly".
The announcement on Wednesday surprised some officials on Capitol Hill, who noted that no drafts of the order had been discussed with Congress prior to the signing.
Fifty-five days before the November 6 midterm congressional elections, officials said they have are seeing some attempts to meddle, from Russia, China and other countries.
Congress is also considering several pieces of legislation that would punish foreign countries for interfering in US elections. "So I think this order is a further demonstration of that", he added. Trump has called the federal investigation into Russia's role in the 2016 presidential election a "witch hunt".
The White House has worked to push back against accusations that Trump was not seriously committed to aggressively protecting USA elections from interference, especially after his press conference with Russian President Vladimir Putin in July.
Trump has also said he accepts the strong consensus view of US intelligence agencies that Russian Federation did interfere, including through propaganda and falsehoods spread on social media.
"President Putin was extremely strong and powerful in his denial today".
Bolton said Wednesday that the White House is open to ideas and proposals from lawmakers, but said new legislation might be slow in coming.
United States lawmakers have introduced various pieces of Russia-related legislation, including the "Deter Act", to set out punishments for election meddling, and what one lawmaker called a sanctions bill "from hell" to punish Moscow for cyber crime and its activities in Syria, Ukraine and elsewhere.
Bolton said the order is not focused on any particular country, because threats to elections come from many different countries and entities. The State Department and Treasury could decide to take additional measures.
Russian Federation has continually denied allegations of interfering in the campaign.
The executive order could blunt momentum in Congress for legislation that would impose harsher and more direct penalties on Russian Federation for any future election meddling. The order, according to administration officials, is broad in terms of who and what can be sanctioned.
But Senate Intelligence Committee Chairman Republican Richard Burr expressed hope the new executive order will "send a clear message" to Russia, Iran and others.
The president appeared to side with Mr. Putin and against US intelligence agencies that said the Kremlin meddled in 2016 to sow division, hurt Democrat Hillary Clinton and help Mr. Trump.
"We have seen signs of [meddling from] not just Russian Federation, but from China ... from Iran and even North Korea", he said.
"We have not seen the intensity of what happened in 2016 [but] it's only a keyboard click away so we are taking nothing for granted", he said.
Mr. Trump later said he misspoke and vouched for intelligence agencies' findings.
"This clearly is a process put in place to try to assure that we are doing every possible thing we can, first of all, to prevent any interference with our elections, to report on anything we see between now and the election, but then to do a full assessment after the election to assure the American people just exactly what may have happened or may not have happened", Coats summarized. He just said it's not Russian Federation.