The Law and Justice (PiS) party came first in the Oct. 15 election but lost its absolute majority and is unlikely to find a coalition partner.
Three pro-European parties say they are ready to form a cabinet led by Tusk and have urged President Andrzej Duda not to delay his appointment.
Duda said he would convene the first sitting of the newly elected parliament on Nov. 13, but did not announce whom he would ask to form a new government.
"If the president quickly acknowledges the reality that there is a majority in parliament and allows it to create a government, I guarantee that it will mean a quick payment of funds to Poland," Tusk told reporters in Brussels, where he has been meeting European Union leaders in a bid to unblock funding.
However, speaking before an EU summit in Brussels, Prime Minister Mateusz Morawiecki said PiS "has the largest number of lawmakers ... we are a party that has won the elections for the third time."
At stake is Poland's access to 35.4 billion euros ($37.33 billion) in grants and loans from the EU's COVID-19 recovery fund, which the bloc has suspended until Warsaw rolls back reforms critics say undermine the independence of its judiciary system.
The same concerns are blocking Poland's access to 76.5 billion euros of EU cohesion funds, meant to raise the standard of living in poorer regions.
Access to the cohesion funds can be unblocked at any time as part of the EU's seven-year budget.
Poland has requested adjustments to the amount of money it receives in COVID-19 recovery funds which brings the total to nearly 60 billion euros. These additional funds could be lost if an agreement on them is not reached by the end of this year.
Duda, a PiS ally, said the preliminary date for the lower house's session was the soonest this could be done. He did not say who he would appoint as prime minister, with both Tusk and Morawiecki claiming the role.
Duda has 30 days from election day to convene the first session of the new parliament and then 14 days to nominate a candidate for prime minister.
This article was produced by Reuters news agency. It has not been edited by Global South World.