The move comes a week after Kazimierz Marcinkiewicz resigned from the post following reports of a rift with Jaroslaw Kaczynski.
The development makes the Kaczynski brothers the world's only twins to hold the two highest posts in any country.
Mr Kaczynski's conservative government has to be approved in parliament. The vote is expected next week.
Jaroslaw Kaczynski leads the centre-right Law and Justice party, which won parliamentary elections last September.
At the time, he said it would not be right for the brothers to take the top two government posts in Poland.
Mr Marcinkiewicz is expected to run for mayor of Warsaw later this year, a job previously held by Lech Kaczynski.
Over recent weeks, there had been frequent reports of a rift between Mr Marcinkiewicz and Jaroslaw Kaczynski over economic policy.
'Potato' row
The Kaczynski brothers - opposition activists during communist times - struck out on their own in 2001, when they founded the Law and Justice Party.
Last September, Law and Justice came first in parliamentary elections.
In May, it formed a controversial coalition government with two smaller Eurosceptic parties.
The BBC's Central Europe analyst Jan Repa says that in Paris, Berlin and Brussels, the Kaczynskis tend to be seen as prickly, unsophisticated, provincial nationalists.
Last week, President Kaczynski cancelled a summit meeting with French and German leaders at short notice, claiming illness.
Reports suggested, though, that he may have been upset by a satirical article in a German newspaper, Die Tageszeitung, comparing him to a potato.
Eight former Polish foreign ministers said Mr Kaczynski's failure to give a credible reason for not taking part in the summit had shown contempt for Poland's partners.
At home, the Kaczynskis claim to speak for the ordinary, honest Pole, tired of being bullied and patronised by foreigners and of being ruled by corrupt cliques and incompetent bureaucrats, Jan Repa says.
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