The most famous example is the Dred Scott v. Sandford decision, which I have written about in this sub before here.

To reiterate, the case was first argued in February 1856. After the arguments were heard, there was speculation in the press (notably in the anti-slavery New York Tribune on February 20th) about what the justices were going to do. This, though, seemed to be based on public knowledge of the justices' track records, along with (probably) some insider knowledge of what each justice's plan was. But no real specifics.

But a little over a week later, on February 29, 1856, the Tribune ran another story, which appeared to be a direct leak of conversations about the planned ruling among the nine Supreme Court Justices. In part because of this, in May, the Court took the extraordinary step to not issue a ruling at all at the time, and instead announced they would be re-hearing the case in December.

By December, James Buchanan had been elected president, and he lobbied his personal friend on the court, John Catron, to convince the court to issue a broader ruling than what the initial leak had suggested. A "broad ruling" meant that the court would address the Constitutionality of the Missouri Compromise of 1820, which was not particularly central to the case. Not only that, but Buchanan also lobbied Robert Cooper Grier, a Northern Democrat, to join the majority opinion, so that it wouldn't be seen as "sectional", i.e., a partisan North-vs.-South decision.

Buchanan got his wish, and the decision was leaked to him. I don't think he saw the actual text of the decision before it was issued, but Catron (Edit: and Grier) wrote to him, basically telling him what it was going to be.

So, in his Inaugural Address on March 4, 1857, Buchanan made a statement that hinted he knew what was coming, saying that the future of slavery in the territories was a "pending" question before the Supreme Court that "will, it is understood, be speedily and finally settled".

What was he talking about? Two days later, the public got their answer, when the Dred Scott decision was issued. This raised a lot of questions about what the President knew and how did he know it. /u/no-tea has written before in this sub about the ethical, and perhaps impeachable, impropriety of Buchanan's actions in the Dred Scott case.

More recently, there were questions about an improper relationship between Supreme Court Justice Abe Fortas and the president who nominated him, Lyndon Johnson. Fortas had been a political operative for Johnson in his early career, and the two remained close, talking on the phone just about every day when Johnson was president and Fortas was on the Supreme Court. While someone with more knowledge of the situation might be able to offer a more specific answer, I don't think there's much doubt that Fortas was feeding at least some knowledge to LBJ about the private discussions among the court.

It got to the point that the Senate Judiciary Committee even investigated Fortas over this impropriety, and other ethical breaches he had made. This ultimately led to Chief Justice Earl Warren asking Fortas to resign in 1969, which he did.