The directive was given by Justice N K Modi on a writ petition filed the son of Khatun Bi, the missing woman.
The court asked the government to file a report on steps taken so far to trace the 68-year-old woman.
Khatun Bi went to Ajmer by a special train under the "Mukhya Mantri Teerth Darshan Yojna" scheme on September 13 from Alot town in Ujjain district and visited the dargah of Khwaja Moinuddin Chishti in the Rajasthan town on the following day.
Later, her family members filed a missing person complaint with the Ajmer Dargah Police Station.
Despite writing several letters to authorities, they remained insensitive and have not taken any steps to locate Khatun Bi, Bhatnagar alleged.
Under the scheme, state government makes all arrangements, including boarding, lodging and local transport, for pilgrims free of cost.