NEW YORK (Reuters) - Prosecutors in Donald Trump's criminal trial are expected to call more witnesses on Tuesday as they build out their argument that he was responsible for illegally covering up a hush money payment to a porn star in the run-up to the 2016 election.

Prosecutors have shown the former president's signature was on payments at the heart of the case. Over the next two weeks they aim to demonstrate that Trump, running again for president, was responsible then for an illegal cover-up.

On Monday, jurors saw the 34 business records that prosecutors say were falsified by Trump to obscure his reimbursement of then-lawyer Michael Cohen, who made a $130,000 payment to porn star Stormy Daniels to keep quiet about a 2006 sexual encounter she says she had with Trump.

The first former U.S. president to undergo a criminal trial, Trump has pleaded not guilty and denies that he ever had sex with Daniels, whose real name is Stephanie Clifford.

A former Trump employee testified he had been told by Trump's top financial officer that the reimbursements to Cohen were for expenses incurred during the campaign. That could undercut an argument made by Trump's lawyers that the payments were for legal work.

However, neither that employee nor another who testified on Monday was able to say whether Trump himself directed the paper trail to be falsified to hide the payments to Cohen - a hole that prosecutors will try to fill with additional testimony.

Jurors have yet to hear from Cohen or Daniels, whose real name is Stephanie Clifford.

They also have yet to hear from Karen McDougal, a former Playboy model who was paid $150,000 during the campaign by the National Enquirer for her story of an alleged affair with Trump in 2006 and 2007.

The tabloid's former publisher, David Pecker, has testified that the paper never ran McDougal's account, due to a "catch and kill" agreement with Trump to bury stories that could have damaged his 2016 presidential bid.

Pecker was targeted in a "swatting" incident, meant to trigger a potentially dangerous response by law enforcement, the same day he took the witness stand, according to police records seen by Reuters.

Trump says the trial is a politically motivated attempt to undercut his campaign to win back the White House from Democratic President Joe Biden in the coming Nov. 5 election.

Justice Juan Merchan has fined Trump a total of $10,000 and warned Trump he could be jailed for violating a gag order that bars him from making public statements about jurors, witnesses and the family members of either the prosecutors or the judge himself if meant to interfere with the case.

The case is widely seen as less consequential than three other criminal prosecutions Trump faces, but it is the only one certain to go to trial before the election.

The other cases charge Trump with trying to overturn his 2020 presidential defeat and mishandling classified documents after leaving office. Trump has pleaded not guilty to all three.

(Reporting by Jack Queen and Luc Cohen in New York and Andy Sullivan in Washington; Editing by Howard Goller)

By Jack Queen, Luc Cohen and Andy Sullivan