Congressional committees, along with the FBI, are investigating what U.S. intelligence agencies have concluded were attempts by Russia to influence the Nov. 8 presidential election in Trump's favor. They also are looking at any links between Russia and Trump.
Schiff said Trump himself tried to shift Congress' focus away from that core mission of foreign intervention.
"I think his tweets tell the story," Schiff said on CNN's "State of the Union." "And the story is look over there - at leaks, and look over there - at anything the Obama administration we can claim did wrong on incidental collection or anything else."
"But whatever you do, under no circumstances look here, at me or at Russia," he said.
Trump has repeatedly used Twitter to attack reports on Russian election meddling as "fake news" and "witch hunts" and denounce leakers of classified information on the issue.
"The real story turns out to be SURVEILLANCE and LEAKING! Find the leakers," Trump said in a Twitter post on Sunday.
Republican Senator John Cornyn, a member of the Senate Intelligence Committee, was asked about Trump's Twitter commentary on the congressional investigations.
"Sometimes I think this is a distraction from what we should be doing," Cornyn said on CBS' "Face the Nation."
The Senate panel intends to begin interviewing as many as 20 people, including Jared Kushner, Trump's son-in-law and one of his closest advisers, as early as Monday.