Szymon Hołownia was speaking at a joint news conference with Tusk, after the two held talks earlier in the day, Polish state news agency PAP reported.
Hołownia, the Speaker of the Sejm, Poland's lower house, said that "time permitting," MPs "will elect a new prime minister on December 11, or else on December 12, in line with the voters' verdict from October 15."
Earlier on December 11, conservative Prime Minister Mateusz Morawiecki is set to deliver a policy speech in parliament, followed by a floor debate and a vote of confidence at around 3 p.m.
Morawiecki was reappointed prime minister by President Andrzej Duda last week, but his conservative Law and Justice (PiS) party lacks a majority in the new parliament.
If Morawiecki's Cabinet fails to secure a vote of confidence from MPs, the lower house will be constitutionally obliged to appoint another prime minister, reporters were told.
A bloc of opposition parties had put forward Tusk as their candidate for head of government.
At Friday's news conference, Tusk said that MPs would elect a new prime minister "on December 11."
He added that his policy speech as the new prime minister, and the vote of confidence in his Cabinet, "will be held on December 11, if time permits, or on December 12."
Tusk said the opposition parties "expect the president to swear in the new government on December 13."
Polish voters elected a new set of 460 MPs and 100 senators when they went to the ballot box on October 15.
Morawiecki's right-wing Law and Justice party won Poland's October 15 election, but lost its parliamentary majority and looks unlikely to stay in power for a third consecutive term.
A bloc of pro-EU opposition parties in mid-November secured a resounding victory in Poland's new parliament during its first key vote to elect a house Speaker.
Poland's opposition groups have signed an agreement to form a coalition government, pledging to restore the rule of law, promote green energy and relax abortion rules, among other policies.
Opposition leader Donald Tusk, a former top EU official, has already chosen most candidates for ministerial roles in his future Cabinet, according to the Rzeczpospolita newspaper.
Hołownia told reporters last Friday that "all the names of the members of Donald Tusk's future government have been agreed on."
Tusk was Poland's prime minister from 2007 to 2014.
Source: IAR, PAP