The dispute pits Tusk's centrist government against conservative President Karol Nawrocki over efforts to reverse a judicial overhaul carried out by the right-wing Law and Justice (PIS) party between 2015 and 2023.
Those changes drew criticism from European courts over concerns about judicial independence, Politico reported.
Nawrocki, who was elected with PiS backing last year, has refused to swear in some Tribunal judges elected by the current parliamentary majority, while the court's PiS-linked leadership has prevented them from taking office, according to Politico.
The European Commission said it was closely monitoring developments in Poland and, in a carefully worded statement, appeared to support Tusk's position, Politico said.
"The Polish Constitutional Tribunal needs to be able to function in accordance with the Polish constitution and the case law of the Court of Justice of the EU," European Commissioner for Democracy, Justice and the Rule of Law Michael McGrath said, according to Politico.
"This matters for every Polish citizen, and it is for the Polish authorities to ensure that," McGrath added.
He also said the Venice Commission, a Council of Europe advisory body on constitutional law, had been clear that appointing new judges falls within the powers of Poland’s lower house of parliament, the Sejm, Politico reported.
Tusk vowed on Saturday not to back down in the dispute.
"In some issues, there can be no compromise," Tusk said in an address to the nation, comparing calls for reconciliation to a thief demanding to keep half the contents of a stolen purse.
"We have waited too long for the treasure of an independent, democratic Poland to be frightened into retreat," Tusk said.
Tusk also criticised Nawrocki, who turned his plane around mid-flight while en route to Senegal and returned to Warsaw on Friday.
Videos showed Nawrocki pushing his way into the Tribunal shortly after midnight and confronting judges challenging its leadership, Politico said.
"In a state governed by the rule of law, a politician cannot enter a court or tribunal and give orders or intimidate judges, even if he is the president," Tusk said.
Meanwhile, Nawrocki's spokesman, Rafał Leśkiewicz, said on Sunday that the President's Office would notify prosecutors that judges Anna Korwin-Piotrowska, Maciej Berek and Krystian Markiewicz "violated the president's personal integrity."
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Source: Politico