But Senate Democrats and Republicans who have taken a harder line on Russian interference said the executive order fell short and that only legislation could force tough action against Moscow. A sanctions bill passed last year forced the administration to enact two rounds of sanctions this year.

“Something is better than nothing, but I doubt it will be a substitute for legislation,” said Senator Lindsey Graham, Republican of South Carolina, one of the leading Republican sanctions proponents.

Senators Marco Rubio, Republican of Florida, and Chris Van Hollen, Democrat of Maryland, have written a bill, the Deter Act, to impose mandatory sanctions on anyone who attacks the American election.

While the executive order would primarily target the people and entities that attack the election system, lawmakers said, the Senate legislation would have wider economic sanctions targeting financial institutions, oligarchs and others.

In an interview, Mr. Van Hollen said the executive order was merely a version of his legislation “without the teeth.”

“The Russians need to know the response will be certain and severe, and the executive order provides neither of those key elements,” he said. “As I look at this, it seems aimed more at deterring congressional action on the Deter Act than deterring Putin’s interference in our elections.”

Mr. Van Hollen predicted that if the measure got a vote on the Senate floor, it would pass with a comfortable bipartisan majority. But the challenge will be getting the bill to the floor.