Mr Kaczynski's Law and Justice party never accepted an inquiry that found the Smolensk crash was an accident.
The party, which was returned to power in elections last month, accuses Mr Tusk of negligence.
He resigned as prime minister last year to become European Council president.
Government spokesman Elzbieta Witek said Poland's State Tribunal would be "a good thing" for Mr Tusk, referring to a court that handles cases against elected and other senior officials.
She was echoing a call from a government minister, Adam Lipinski, who said the former prime minister had "a lot to answer for" and should be prosecuted after his initial term of office came to an end in Brussels in 2017.
On 10 April 2010, 96 Poles were killed when the Tupolev plane they were travelling in went down in thick fog, short of the runway near Smolensk in western Russia.
The president, his wife and senior government officials were on their way to Katyn to mark the 70th anniversary of the murder of thousands of Polish officers by Soviet secret police.
Separate investigations ruled out any kind of plot, blaming the conditions and poorly trained pilots. Leaked transcripts also indicated the pilots had come under pressure from people on board.
Mr Lipinski, a minister in the prime minister's office, told Polish newspaper Super Express that Mr Tusk had "given away" the Smolensk investigation to the Russians, had been negligent in explaining the disaster and had failed to bring back the wreckage to Poland.
The Law and Justice party is still run by the late president's brother and identical twin, Jaroslaw Kaczynski, although the prime minister is Beata Szydlo.
The government spokeswoman said there was no plan to seek a prosecution against Mr Tusk, insisting it was her private view.
But ministers this week shut down the website of the Polish state's Smolensk crash investigation.
A Polish government spokeswoman has called for former Prime Minister Donald Tusk to be put on trial for his handling of the 2010 air disaster in which President Lech Kaczynski died.