After Michael Cohen struck a new plea deal with the special counsel Robert Mueller, President Donald Trump and his lawyers slammed his former fixer as a "liar" and said "we can't trust him."

At the same time, they said Trump's version of events about a defunct effort to build a Trump Tower in Moscow during the election lines up with what Cohen told prosecutors.

A source close to Trump's legal team told INSIDER the president's lawyers were furious that Mueller waited to announce Cohen's plea deal until after he had secured Trump's written answers to questions about collusion.

The source said Trump's lawyers believe the timing of the event indicates Mueller is trying to trap Trump into perjuring himself, by getting his answers first and then presenting him with evidence that contradicts what he said.

President Donald Trump and his lawyers can't seem to get their story straight on whether his former lawyer and longtime fixer, Michael Cohen, is a liar.

On the one hand, they have repeatedly said Cohen cannot be trusted to tell the truth. On the other, they said Trump's story lines up with Cohen's as it relates to the Trump Organization's effort to build a Trump Tower in Moscow at the height of the 2016 election.

On Thursday, Cohen pleaded guilty to one count of making false statements to Congress about his involvement in the project.

Read more: Michael Cohen's latest plea deal shows Mueller's 'starting to finally get to the truth' behind Trump and Russia

Prosecutors said Cohen knowingly misled congressional investigators when he said negotiations for the deal ended in January 2016 and that he did not discuss it extensively with Trump Organization executives, including Trump and his family members.

Trump unleashed a tirade against Cohen after his plea deal was announced, saying, "So very simply, Michael Cohen is lying."

Trump's lead defense lawyer, Rudy Giuliani, struck a similar tone.

"He has so many different versions of the same stories, so by definition he is a liar and we can't trust him," Giuliani said of Cohen. "He has lied, so how can we believe him?"

But at the same time, Giuliani told The New York Times that Trump's story is the same as Cohen's when it comes to the Trump Tower Moscow deal.

The president knew about the deal and discussed it with Cohen before it fell apart, Trump's lawyers told The Times. "The president said there was a proposal, it was discussed with Cohen, there was a nonbinding letter of intent, and it didn't go beyond that," Giuliani said.

Prosecutors are said to have asked Trump about the defunct project in a series of questions that Mueller's office sent over to the president. Trump's lawyers sent back Trump's written answers to the questions last week.

A source close to Trump's legal team told INSIDER the president's lawyers were furious that Mueller waited to announce Cohen's plea deal until after he had secured Trump's written answers.

The source said Trump's lawyers believe the timing of the event indicates Mueller is trying to trap Trump into perjuring himself, by getting his answers first and then presenting him with evidence that contradicts what he said.

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Jeffrey Cramer, a former longtime federal prosecutor who spent 12 years at the Department of Justice, weighed in on that argument.

"It's only a perjury trap if you lie," Cramer told INSIDER. "If you tell the truth, there's no question of a perjury trap. It's telling that everyone in Trump's orbit just seems to automatically assume he's going to lie if given the opportunity."

Prosecutors said in Cohen's charging document that he "discussed the status and progress of the Moscow Project" with Trump "on more than the three occasions Cohen claimed" to the Senate Intelligence Committee and that "he briefed family members" of Trump within the Trump Organization about it.

He was also discussing it with the Russian-born businessman Felix Sater as late as June 2016.

Trump does not face significant criminal liability based on the Trump Tower Moscow deal itself, because it isn't illegal for Americans to do business in Russia.

But Cohen's guilty plea is a confirmation that the president was not being truthful when he denied having any financial interests in Russia during the campaign. Later Thursday, BuzzFeed News dropped a bombshell report that said the Trump Organization wanted to give Russian President Vladimir Putin the $50 million penthouse in Trump Tower Moscow.

The development will likely be of keen interest to prosecutors as they examine Trump's long history of praising Russia and Putin and of siding with Russia against US intelligence agencies.

In other words, the problem for the president arises not from actions but from motive, said Jens David Ohlin, a vice dean at Cornell Law School who is an expert on criminal law.

"Not Cohen's motive, but Trump's motive for deferring to Russia over and over again," Ohlin told INSIDER. "The motive is money and business deals. This gives Mueller the last piece of the puzzle."