Washington (CNN) The Russia "cloud" is growing over President Donald Trump and threatening to wreck another week for the embattled White House as one of his closest allies tries to avoid a public spectacle on Capitol Hill.

Attorney General Jeff Sessions' announcement that he would testify before Senate Russia investigators caught everyone -- including the intelligence committee -- by surprise. Now, before he even testifies, the focus is both on Sessions and whether or not he would appear before the TV cameras like former FBI Director James Comey did last week, when 19.5 million Americans watched.

Trump team has struggled to move the focus away from Russia.

The White House is hoping this week to drive a message focused on his agenda. His daughter and top adviser, Ivanka Trump, is leading events focused on workforce development and college affordability.

But Russia -- and Trump's own tweeting -- threaten to swallow that effort whole, much like last week's largely forgotten "Infrastructure Week." In addition to Sessions' possible testimony, the question remains whether or not Trump taped his conversations with Comey.

The President himself tweeted early Sunday morning touting his economic record.

...way up. Regulations way down. 600,000+ new jobs added. Unemployment down to 4.3%. Business and economic enthusiasm way up- record levels! — Donald J. Trump (@realDonaldTrump) June 11, 2017

"The #FakeNews MSM doesn't report the great economic news since Election Day. #DOW up 16%. #NASDAQ up 19.5%," Trump wrote in the first of two tweets.

Minutes later, Trump demonstrated the White House's messaging problem, immediately shifting to Comey and kicking off another news cycle: "I believe the James Comey leaks will be far more prevalent than anyone ever thought possible. Totally illegal? Very 'cowardly!'"

I believe the James Comey leaks will be far more prevalent than anyone ever thought possible. Totally illegal? Very 'cowardly!' — Donald J. Trump (@realDonaldTrump) June 11, 2017

Adding to the tension for the White House are fast-approaching deadlines in Congress to approve a health care bill before a September 30 deadline and agree on a tax reform plan -- both top campaign promises, which appear to be longshots.

Trump's chief White House liaison to the Hill, Marc Short, admitted last week in a call with reporters that Russia questions had sucked much of the oxygen out of the room

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Sessions: Will he or won't he?

The attorney general is coming off a disastrous week of his own. After the White House wouldn't give Sessions a full-throated public endorsement, Comey told senators that Sessions may have had a third, undisclosed meeting with the Russian ambassador to the US, Sergey Kislyak.

He had been scheduled to answer questions before the House and Senate spending committees Tuesday, but instead wants to show up to the intelligence committee instead. That possibility could mean a closed hearing, unlike the drama of last Thursday with Comey.

In an odd twist for Congress, senior members of the Senate intelligence committee were unsure if Sessions would even be allowed to appear.

"I don't know whether it will happen -- don't know whether it's going to be public," Sen. Dianne Feinstein, a California Democrat, told CNN's Brianna Keilar on "State of the Union" Sunday.

Feinstein, who used to chair the Senate intelligence committee and is the top Democrat on the Senate judiciary committee, tossed a curveball back at the attorney general, arguing he should testify publicly before the seasoned lawyers and prosecutors of the Judiciary Committee.

"I challenge the jurisdiction, to some extent. I'm on both committees, as you know," she said. "There is an opportunity to look at the law with respect to obstruction of justice, to hold a hearing, and also to have those relevant people come before the judiciary committee."

Meanwhile, behind the scenes, Democrats on the Senate intelligence committee believe Sessions is trying to use their venue to duck public scrutiny.

Sen. Ron Wyden, an Oregon Democrat on the committee, who pressed Comey last week on why Sessions' was involved in his firing if he had recused himself, argued that Sessions had no reason to seek a classified briefing.

"The American people also deserve to hear the attorney general's answers to these questions, as well as others related to his meetings with the Russians and his failure to disclose those meetings to the Senate Judiciary Committee," Wyden wrote Sunday in a letter to intelligence panel leaders. "None of this needs be classified."

The test comes as special counsel Robert Mueller appears to be ramping up his probe -- hiring away a handful of seasoned federal criminal investigators that could hint at deeper, more serious legal trouble for Trump and his team.

Meanwhile, Trump's own Russia team, led by Marc Kasowitz, brought on veteran Republican communicator Mark Corallo, a former spokesman to former-Attorney General John Ashcroft who is steeped in crisis communications from his time helping defend Karl Rove in the midst of Valerie Plame case.

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Moving on?

Trump's allies attempted to move the national storyline away from Russia on the Sunday talk shows, arguing that Comey's testimony about former Attorney General Loretta Lynch to call the Hillary Clinton email investigation a "matter" was more compelling.

"I'm calling for an end to the investigations about the President Trump's campaign colluding with the Russians. There's been no evidence of it. I don't think that should continue," Republican National Committee Chairwoman Ronna Romney McDaniel said on "Fox News Sunday."

But even Republicans in the Capitol have continued pressing for answers in the Russia probes. Senate judiciary committee Chairman Chuck Grassley and Feinstein formally requested Comey's memos of his meeting with Trump from Comey's friend Daniel Richman.

House Russia investigators, led by Texas Republican Rep. Michael Conaway and California Democratic Rep. Adam Schiff, sent a formal request to the White House for any records of Trump's private meetings with Comey.