He added, "We were requested by the chairman and vice chairman of the intelligence committee to let the supervisors of election in Florida know that the Russians are in their records."
The newspaper reported he said something similar in Tallahassee on Tuesday, but declined to elaborate.
"That's classified," Nelson said Tuesday.
Nelson, who is running to keep his seat against Florida Republican Gov. Rick Scott, did not immediately respond to a request for comment from CNN.
Florida GOP Sen. Marco Rubio's office declined to comment to CNN on the comments made by Nelson.
The Florida Department of State pushed back on the claims by Nelson that Russian operatives have "penetrated" Florida voting systems.
"The Florida Department of State has received zero information from Sen. Nelson or his staff that support his claims," Florida Department of State spokesperson Sarah Revell told CNN. "Additionally, the department has received no information from the US Department of Homeland Security, the Federal Bureau of Investigation or the Florida Department of Law Enforcement that corroborates Senator Nelson's statement and we have no evidence to support these claims. If Sen. Nelson has specific information about threats to our elections, he should share it with election officials in Florida."
Revell also stressed that "state and local election officials have taken significant steps to ensure the security and integrity of our elections."
The Department of Homeland security responded to Nelson's comments by saying their department and officials with Florida "have partnered on a number of initiatives to secure their election systems" but said DHS continues "to assess Russian actors were not able to access vote tallying systems."
"While we are aware of Senator Nelson's recent statements, we have not seen any new compromises by Russian actors of election infrastructure," Homeland Security spokeswoman Sara Sendek said in a statement. "That said, we don't need to wait for a specific threat to be ready."
Nelson's comments come after the two Florida senators sent a letter to 67 county election supervisors about potential threats ahead of the 2018 midterms, asking them to take advantage of the resources provided by the Department of Homeland Security.
"We encourage you in the strongest terms to take advantage of those resources, and to let us know about your experience with DHS and FBI," Nelson and Rubio both wrote in the letter.
A spokeswoman for Senate Intelligence Chairman Richard Burr declined to comment.
The top Democrat on the committee, Sen. Mark Warner of Virginia, told CNN in a statement: "Russian activities continue to pose a threat to the security of our elections, as Sens. Nelson and Rubio rightly pointed out in their letter. As Vice Chair of (the committee), I hope all state and local elections officials, including Florida's, will take this issue seriously."
The statements from Nelson come amid growing attention toward whether the Russians are trying to influence the midterms, such as intelligence officials said they did during the 2016 election.
"We continue to see a pervasive messaging campaign by Russia to try to weaken and divide the United States," he said. "The President has specifically directed us to make the matter of election meddling and securing our election process a top priority."
This story has been updated with additional developments.
CNN's Jeremy Herb and Pamela Kirkland contributed to this report.