Justice Sheikh Najamul Hassan of the Lahore High Court issued the order in response to a petition filed by Gill's father, Sikandar Gill, seeking the "safe recovery" of his daughter.
Gill told the court that his daughter arrived in Pakistan on August 25 to attend an exhibition of diamonds in Lahore.
She went missing and was never seen in the city, he said.
"We made efforts to contact her but failed," Gill said.
He asked the court to direct the police to find his daughter and produce her in court at the earliest.
The judge issued a notice to Lahore Police chief Aslam Tareen and directed him to submit a report by December 31.
Earlier, investigators had said that Rajvindar Kaur Gill had come to Lahore in connection with her "gems business".
She is a distant female relative of Sukhbir Singh Badal, the Deputy Chief Minister of Indian Punjab.
According to a report submitted by investigators, Rajvindar came to Lahore from Canada via Dubai in mid-August.
She stayed in three hotels in Lahore and met some people whom police are yet to trace.
Rajvindar went missing on August 31. On that day, she was to leave for Karachi to meet journalist Iqbal Hussain of News One channel.
Investigators said they quizzed Hussain but could not find more than Rajvindar's plans to meet him.
The Punjab Police chief recommended to the provincial government that a joint investigation team should be formed for a probe.
However, a source in Punjab Police said that the possibility that Rajvindar might be in the custody of intelligence agencies could not be ruled out.
Lahore: A Pakistani court on Friday directed the Lahore Police chief to file a detailed report by December 31 on Rajvindar Kaur Gill, a Canadian woman of Indian origin reported missing in the city since August.
First Published: Friday, December 28, 2012, 21:42