Mr Jaiswal allegedly made the remarks during a meeting of the local businessmen.
A first information report or FIR has been filed against Bihar BJP chief Sanjay Jaiswal and Sweety Singh, the party's candidate for Kishanganj assembly bypoll, for allegedly violating the model code of conduct, an official said.
The complaint was registered at Sadar police station on Saturday evening following a written complaint by Kishanganj Sadar Circle Officer Shafi Ahmad, Sadar Sub-Divisional Magistrate (SDM) Shanawaz Ahmad Niyazi said.
The complaint was filed after a video surfaced on social media purportedly showing Mr Jaiswal exhorting businessmen to close their establishments on the polling day and promising to pay Rs 500 to each of their staff members in their firms and houses, the SDM said.
Mr Jaiswal allegedly made the remarks during a meeting of the local businessmen held on Thursday at the residence of Kishanganj Municipal Council vice chairperson Aanchi Devi.
Further action will be taken against others on the basis of the video, senior police officer Niyazi said.
Bypolls to Kishanganj and four other assembly seats in Bihar will be held on Monday as the incumbent MLAs were elected to parliament in the national elections.