The Twitter officials' offer to report via video calls was rejected by the UP police.
The Uttar Pradesh police have summoned Twitter India Managing Director Manish Maheshwari for questioning on Thursday in connection with tweets on the assault of a elderly Muslim man in Ghaziabad. He had made himself available on video call.
The police have also asked Twitter India's Resident Grievance Officer Dharmendra Chatur to report to them on the same day. Their notice says these two officers were responsible for the activities on the microblogging platform.
"You couldn't remove certain tweets even after you were asked to by authorities. You understand Indian laws and are bound to follow them," the notice said.
A case was filed last week against the social media site, along with several journalists and Congress leaders, alleging that that the tweets they had shared on the assault were "misleading" and aimed at "provoking communal sentiments".
Earlier, Ghaziabad resident Abdul Samad had alleged that he was thrashed by some men and forced to chant "Jai Shri Ram" and "Vande Mataram".
The police, however, denied any "communal angle" in the case, claiming he was beaten up over the amulets he sold. They alleged he was attacked by six persons -- Hindu and Muslim -- known to him.
Mr Samad's family denied the police's claims.
"The police is wrong in saying that my father used to sell tabeez (amulets). No one in our family does this business. We are carpenters. The police are not saying the right thing -- let them investigate and prove it," the assault victim's son Babloo Saifi told NDTV.
On Friday, Mr Maheshwari was asked to report within seven days at the Loni Border police station near Ghaziabad. The Twitter India chief replied saying "does not deal with the case directly".
This is the first time a case has been filed against the social media giant after the Centre's new rules for online news publishing platforms came into effect.
Union Minister Ravi Shankar Prasad had last week said, "What happened in UP was illustrative of Twitter's arbitrariness in fighting fake news...While Twitter has been over enthusiastic about its fact-checking mechanism, it's failure to act in multiple cases like UP is perplexing and indicates its inconsistency in fighting misinformation."
The Editors Guild of India had earlier condemned the filing of cases against news website The Wire and the journalists for their tweets. It was "deeply concerned by the UP Police's track record of filing FIRs against journalists to deter them from reporting serious incidents without fear of reprisals".