Babis said after a meeting on Thursday the president remained unready to act. The Czech Social Democratic Party has said the removal of the culture minister they had originally nominated is a condition for it to stay in the governing coalition.
The Social Democrats asked for the replacement of the minister, Antonin Stanek, after he fired two directors of the state run art galleries, controversial moves in April that were met with a wave of protests.
Replacing a minister is generally a routine matter and the Czech Constitution says the president should comply with such requests from the prime minister. Zeman has not made fully clear why he is not acting on the request, but has said the Constitution does not give him a deadline for doing so. The president has also expressed his doubts about whether the proposed new minister, Michal Smarda, can do the job.
The Social Democrats plan to decide on Monday whether to pull out. If it does, several options are open, including an early election. The party's chairman, Interior Minister Jan Hamacek, is set to meet with Zeman on Friday.