In a tweet, Surat resident Babu Subramaniam Thottapilly posted a photograph of his son Sandeep Thottapilly, his daughter-in-law and their two children, and urged Ms Swaraj to take the matter up with the authorities in the United States.
Sandeep (42), his wife Soumya (38), son Siddhant (12) and daughter Saachi (9) went missing in the US state of California on Thursday while travelling from Portland to San Jose.
The authorities in the US fear the worst for an SUV, matching the missing family's vehicle, was believed to have been swept away by a swollen river, according to a media report in the US.
Sandeep worked as a vice-president with the Union Bank and lived in Los Angeles with his family.
The banker's father, originally from Kerala and living in Gujarat's Surat for several years, was in a state of shock and unwilling to speak to PTI.
Sandeep grew up in the Gujarat city and settled down in the US over 15 years ago, said a relative when contacted by PTI over the phone.
"Till now we have not received any information about the family's whereabouts from the Indian embassy in the US," the relative said.
Babu Thottapilly had earlier tweeted, "Respected Sushma Swaraj ji, I am a senior citizen and resident of Surat, Gujarat. My son Sandeep Thottapilly is residing in California, USA. He and his family are reported missing since last Thursday.
Along with the message, he posted a photograph of his son and his family.
"We have taken help of Facebook post to reach out to more and more people in California...," he said.
A post by Babu Thottapilly on Union Bank's Facebook page read: "This is to inform you that my son and his family have been missing during the trip to Oregon. The complaint in this regard has been registered with the police authorities and they are looking for his whereabouts.
"Till that time I would like to inform you that kindly do the needful from your end to ensure that law authorities are able to track him and his family."
A media report in the US said authorities were looking for a maroon Honda Pilot that went down an embankment on Friday at around 1:10 pm on Highway 101 near the Dora Creek.
The vehicle is similar to the one the Thottapilly family was travelling in. They were on their way back home in the southern California city of Valencia from Portland, Oregon, after a road trip. They were to have stopped by in San Jose to meet their relatives but never turned up, The Mercury News reported.