The dispute deepens a standoff between parliament and the presidency over the composition of the court.
Maciej Berek, who was elected to the tribunal by the Sejm (lower house of parliament) on 4 September, said in a letter to the president this week that he was asking for an "immediate" date and venue for his swearing-in.
Under Polish law, failure to take the oath can be treated as a refusal of the post.
Speaking to private broadcaster TVN24, Berek said taking the oath was his legal duty following his election by parliament, and that he would ensure it happened "in whatever way the president decides to participate in the process".
He added he would wait "a few short weeks" for a response from the presidential chancellery before considering other steps.
He is the eighth judge elected by the current parliament to face a dispute over his swearing-in.
Nawrocki has declined to formally administer the oath to four of the seven judges elected previously, citing procedural objections raised by his office.
One such concern was that parliament had failed to specify which seats the judges were filling or when their terms began.
The four judges instead took their oaths "before the president" at a ceremony in the Sejm, which the presidential chancellery does not recognise as valid.
The tribunal's president, Bogdan Święczkowski, has barred them from hearing cases.
The four judges have lodged a complaint with the European Court of Human Rights.
Poland's national prosecutor's office is also investigating whether the president's office improperly facilitated withholding their oaths and preventing them from taking part in hearings.
The presidential chancellery has separately questioned whether Berek meets the formal requirement of 10 years' experience as a legal counsel.
(ał)
Source: PAP