Kolaras and Mungaoli have 2,44,457 and 1,91,009 eligible voters, respectively.
Counting of votes would be undertaken on February 28.
The assembly seats fell vacant following the death of sitting Congress MLAs Mahendra Singh Kalukheda (Mungaoli) and Ram Singh Yadav (Kolaras).
Bharatiya Janata Party's Baisaab Yadav and Brajendra Singh Yadav (Congress) were locked in a direct combat in Mungaoli.
Both Chouhan and Scindia have led hectic campaigns for their parties in these bypolls.
During the campaigning phase, the Congress had made several complaints to the Election Commission of India alleging violation of the Model Code of Conduct by the BJP.
In its order yesterday, the Commission stated the Chief Minister violated the poll code by indirectly announcing the construction of bridge over a river between Kukreta and Buxanpur. While the turnout is Kolaras was 70%, in Mungaoli it was 77%.
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The electronic voting machine at booth number 57 in Kolaras did not work in the morning, said.
The BJP nominees are Devendra Jain from Kolaras and Baisaab Yadav from Mungaoli.
"No one would like to be in my position today". BJP never believes in politics of pressure. "Still the matter was flashed by the media in a biased manner".
Polling began at 8 am and would continue till 5 pm, the official said, reports PTI.
A win in any of the assembly constituencies would bolster support for Chouhan, who lost to Congress with its candidate Nilanshu Chaturvedi winning with 14,133 votes in the Chitrakoot assembly by-election.
The BJP deployed Union ministers Dharmendra Pradhan and Jual Oram to the constituency while Smriti Irani addressed three rallies, highlighting the importance the saffron party is giving to the bypoll. A resident of the town told ANI that voting had not begun there due to the faulty machine, and that he had been standing in the queue for an hour.
The technical snags were rectified in these booths and voting continued. The ruling BJP has been in power since 2008.
The results of these bypolls will set the mood for Assembly elections in Madhya Pradesh.
He was, however, defeated in 1995 while contesting as an independent candidate and 2004 Assembly elections as BJD candidate from the constituency.