President Trump's decision to call his executive order on immigration a "travel ban" on Monday morning has the potential to alter the Supreme Court's calculus as it reviews the travel ban litigation speeding toward the high court, according to legal experts.

The president used Twitter on Monday morning to argue that the Justice Department ought to have stuck with his first executive order seeking to implement the travel ban, and said the second version was "watered down." But both have been struck down by federal judges, and the Trump administration last week asked the Supreme Court to review an appeals court decision against the order.

Part of those decisions were based on arguments that Trump made when he was a candidate that the U.S. should ban the entry of all Muslims in order to shore up U.S. national security. Trump's newest tweets made it clear he does see it as a travel ban, despite claims from other White House officials that it is not a travel ban, and instead is a process for enhanced vetting of potential immigrants.

Trump's opponents immediately celebrated the tweets as helpful to their cause. Former acting solicitor general Neal Katyal, who argued against Trump's ban before the 9th Circuit, appeared giddy on Twitter following news of Trump's tweets on Monday.

He said it's "odd" to have Trump acting as his co-counsel. "We don't need the help but will take it!"

The Justice Department had no comment on the president's tweets as of early Monday afternoon. Throughout its arguments before federal courts, the Trump administration has urged federal judges not to read into what the president's motivation behind the order might be.

But even some of Trump's allies say his newest tweets make it harder for the Justice Department to make this argument in the Supreme Court. George Conway, husband of Trump adviser Kellyanne Conway who recently withdrew himself from consideration for a Justice Department position, criticized Trump's tweets as detrimental to his own cause.

He said the tweet "won't help" the administration win the five votes needed in the Supreme Court.

Trump might be the "worst client" the solicitor general has ever had and is his own worst enemy, tweeted Josh Blackman, a South Texas College of Law professor who thinks the odds favor Trump on the travel ban at the Supreme Court.

In an analysis of the potential fallout from the president's tweets, Blackman wrote that Trump's tweets "show utter disregard for the Justice Department's legal strategy."

"According to the Solicitor General, the 90-day ban on entry from certain countries was never meant to be a permanent policy. Rather, it was a pause designed to provide the government an opportunity to reassess its vetting procedures," Blackman wrote. "By insisting on calling the policy a 'travel ban' — notwithstanding his attorney's insistence to the courts that this is not what the policy about — the President undermines the Solicitor General's arguments about the nature of the policy."

"In general, talkative clients pose distinct difficulties for attorneys, as statements outside the court can frustrate strategies inside the court," he added. "These difficulties are amplified exponentially when the client is the President of the United States, and he continuously sabotages his lawyers, who are struggling to defend his policies in an already-hostile arena. I do not envy the Solicitor General's office."

Others think the president's newest tweets may make the Supreme Court avoid hearing the merits of the ban altogether. Brian Goldman, a former clerk to Justice Sonia Sotomayor and attorney in Orrick's Supreme Court and appellate practice, tweeted that Trump's Monday tweets make him less confident that the high court will choose to grant the case.

"[The] tweets are new, relevant evidence, but not part of the record in the case's interlocutory posture," Goldman tweeted.

"[The Supreme] Court may realize decision on the merits is a lose-lose," he continued. "Affirm, and the President will escalate war on the Judiciary. Reverse, and... the Pres. and others will act as if he's brought the courts to heel."

Given all of the pitfalls associated with the president's Monday tweets on the controversial travel ban and his administration's legal strategy, it's unclear why Trump chose to speak out publicly. MSNBC producer Kyle Griffin tweeted that Trump's tweets on the topic came shortly after a segment on an MSNBC morning show criticizing the ban.

The Supreme Court has ordered respondents to file a brief alongside the Justice Department's petition regarding the travel ban litigation by next Monday at 3 p.m. Whether the president's latest tweets on the travel ban change the legal strategy of either side remains to be seen.