The Likud wrote the President's Residence on Tuesday, questioning whether Yamina leader Naftali Bennett could be permitted to form a government during the mandate of Yesh Atid leader Yair Lapid.
Prime Minister Benjamin Netanyahu's party also asked President Reuven Rivlin's office to require all coalition agreements to be submitted when Lapid tells the president he can form a government.
Both requests are expected to be dismissed.
Yisrael Beytenu leader Avigdor Liberman mocked the Likud, noting that Netanyahu himself offered a rotation of three party leaders as prime minister on Sunday.
"This is either a joke or just shameful," Liberman said.