In the letter, Paswan said that the incident which happened in the state guest house cannot be termed a "normal one."
"This can be the result of either gross negligence or some conspiracy. I am of the view that there should be a high-level probe in this incident," Paswan, who heads the NDA ally in Bihar Lok Janshakti Party, said in the letter.
He requested the Home Minister to direct the state government to hold a comprehensive probe into the matter.
On August 21, Shah and Bihar BJP in-charge Bhupendra Yadav had been stuck in the state guest house lift in Patna for about 40 minutes after it stopped a little short of the first-floor alignment level.
While the state BJP called it "negligence" on the part of the state government, the Bihar government said the lift stopped because the weight limit of 340 kg was exceeded. There were four people in the lift -- Shah, Bhupendra Yadav, another senior BJP leader Saudan Singh and RSS leader Nagendraji.