NCP, which wants the Chairman's post, does not have a majority in the Council and would need votes of eight outside members.
The motion was read out by Deputy Chairman Vasant Davkhare (who is from NCP) and 24 of the 28 NCP members stood up to support it.
As per the rules, the motion can be discussed within seven days.
Deshmukh has stayed away from the business of the house since Tuesday and is unlikely to attend the session until his fate becomes clear.
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He said he will fix the day for discussion on the motion after consulting the leader of the House (Eknath Khadse, BJP), the opposition leader (Dhananjay Munde, NCP) and floor leaders of other parties.
Congress members requested that it should not be taken up before Friday.
"Give us two days to discuss the issue within the party," said the senior Congress member Sharad Ranpise.
Meanwhile, NCP claimed it had the support of 32 members, including its own 28 members and a PWP MLC.
Whereas a Congress leader said it sought time as state Congress president Ashok Chavan is in Delhi, attending the Parliament session, and will return tomorrow.