The Washington Post report of the probe of the president himself follows the Trump floating the possibility of firing the new head of the investigation, Robert Mueller, the Justice Department special counsel appointed after the firing of FBI Director James Comey.
The report comes a day after Attorney General Jeff Sessions, a close aide of Trump, vehemently denied any role in Russia's alleged meddling in the 2016 USA polls and dismissed accusations that he discussed such an effort with Russian officials as an "appalling and detestable lie".
"I think he is considering perhaps terminating the special counsel", Ruddy said.
The Senate panel also plans to hear at some point from Jared Kushner, Trump's son-in-law and one of his closest White House aides. Sessions has recused himself from the Russian probe, a move that gave Rosenstein the authority to appoint a special counsel.
Comey told the committee that Trump encouraged him to drop an investigation into former national security adviser Michael Flynn, a claim the president has denied.
Brexit: UK and European Union negotiations to start on Monday
European leaders' biggest concern, however, is that Britain's political chaos is wasting time that could be used to negotiate. British officials travel to Brussels on Thursday ahead of the start of formal talks on Monday.
Quoting officials, the newspaper said one event of interest to Mueller is an exchange on March 22, when Coats told associates that Trump had asked him to intervene with Comey to get him to back off the focus on Trump's former national security advisor Mike Flynn as part of the Federal Bureau of Investigation probe of the Russian Federation affair.
The special counsel does not operate under the purview of the FBI, but took over the investigation previously led by the agency.
For many, the prospect of Mueller's firing immediately brought to mind the 1973 dismissal of special prosecutor Archibald Cox during the Watergate scandal.
"He's got all these weird relationships where he says, "I haven't talked to so-and-so in six months" and then you find out he talked to so-and-so that day", Scarborough said about Trump. Dianne Feinstein of California has said there should be further investigation.
Rosenstein may be asked to address the issue when he speaks at a Senate subcommittee hearing Tuesday morning.
In addition, the House Intelligence Committee has written to Comey to request any notes or memoranda in his possession regarding conversations with Trump be turned over by June 23.
But Thursday's comments come less than 24 hours after the President urged unity after the shooting.