The Senate intelligence committee also hopes to interview Trump Jr. before the end of the year.
"Did you have contact with Glenn Simpson on June 8, 9, or 10, 2016?" reads one of the 94 questions posed to Veselnitskaya by the Senate Judiciary Committee.
The publicist could bear testimony "as early as next week", CNN reported Tuesday.
Simpson was interviewed by HPSCI on November 14, mostly about his involvement with the dossier.
The 51 pages of written answers were submitted to the Senate Judiciary Committee, according to NBC News. But in once case, he asked Trump Jr.to share a link about emails presumed to be hacked from Clinton campaign chair John Podesta's email account, and Trump Jr. did so.
There were also conflicting reports as to whether Wednesday's panel would interview Felix Sater, a Russia-born American real estate developer involved in Bayrock Group LLC, the New York City real estate conglomerate.
Other lawmakers, Republicans as well as Democrats, say the goal of their investigation is to guarantee the integrity of US elections, not to target Trump and his associates.
Trump Jr. was subpoenaed earlier this year by the committee, but it was later dismissed after he agreed to cooperate.
Veselnitskaya has previously given jumbled and inconsistent statements about the events surrounding the Trump Tower meeting. Veselnitskaya met with Russian President Vladimir Putin's prosecutor general, Yuri Chaika, ahead of the meeting to discuss the talking points, The Times reported in October. The person who helped set up that meeting, British music publicist Rob Goldstone, is scheduled to speak to the committee next week.
McFarland Lied to Senate Committee on Flynn and the Russians
Repeated attempts to reach Ms McFarland, who left her post as deputy national security adviser in May, were unsuccessful. Flynn then briefed senior transition team members about his discussions with Mr Kislyak, the records show.
Once it became apparent that she did not have meaningful information about Clinton, Trump seemed to lose interest, Veselnitskaya said, and the meeting petered out...
His lawyer did not comment to NBC.
At issue are recent revelations about communications between Trump Jr. and Wikileaks, the secretive web organization that released hacked Democratic Party emails as part of what US intelligence officials say was a Russian effort to help sway the election toward Trump. Agalarov allegedly proposed the meeting between Trump's son and Veselnitskaya, who had reportedly claimed that she had information aboutthe illegal activities of former Democratic Party presidential candidate Hillary Clinton.
"I would think it's something more than a fishing expedition", says Edwin Truman, a former U.S. Treasury Department assistant secretary for global affairs. "The meeting, essentially, ended there", Veselnitskaya writes.
"I love it", Trump Jr. responded to Goldstone. But Trump Jr. and the White House have denied that allegation, saying that he would have been a fool to turn down an offer of useful information about a political opponent.
Even if he did lose interest in the meeting, Veselnitskaya has also said that Trump Jr. suggested the USA would re-examine sanctions against Russian Federation if his father won the election.
Fusion GPS was working on the dossier project at the time for the Clinton campaign and Democratic National Committee. "If Russia laundered money through the Trump Organization, it would be far more compromising than any salacious video and could be used as leverage against Donald Trump and his associates and family".
"At a minimum, they know there's some fish in this pond and they want to know whether they're nice fish or bad fish", Truman, a nonresident fellow of the Peterson Institute for International Affairs, told VOA.
Also present at the meeting were Trump's son-in-law and his then-campaign manager, respectively.