Election to fill up 55 seats in the Upper House of Parliament was to be held on March 26, but 37 candidates have already been elected without a contest.
The decision on holding the election for the remaining 18 seats will be taken based on the prevailing situation, the sources said.
Nearly 500 coronavirus cases have been reported in India so far, according to Health Ministry data on Tuesday.
As cases of the viral infection surged, authorities have put almost the entire country under lockdown, banning gathering of people and suspending road, rail and air traffic till March 31.
Other Covid-19 developments
In other major development, Prime Minsiter Narendra Modi will address the nation once again at 8 pm tonight.
Delhi Chief Minister Arvind Kejriwal informed that no new cases were reported in New Delhi in last 24 hours and five people were discharged. However, the cases count across India is now every close to 500.