Afterwards, Soubry tweeted: "Thank u everyone for much appreciated support. Voices of hope & tolerance will not be bullied off £Twitter or anywhere else."
On Saturday, the Metropolitan Police said a 25-year-old man had been arrested at his home in Bethnal Green, east London, on "suspicion of sending malicious communications".
Soubry said it was the second incident in three days, having earlier received a death threat via a phone call.
She said: "I think in the current atmosphere, we get so used to all sorts of unpleasantness.
"I think it's the fact it's two in one week that have a very striking similarity. It's certainly shocked me and my staff.
"I have reported it to the police."
Soubry was in London when her constituency office received the threatening phone call. She has reported the matters to Nottinghamshire Police and the parliamentary liaison team based within the Metropolitan Police.
A spokesman said the man was being held in custody at an east London police station. He has not been charged.
The incident comes after Cox's murderer, 43-year-old Thomas Mair, was jailed for life for shooting and stabbing the Labour MP for Batley and Spen outside her constituency office in Birstall, West Yorkshire, in June.
Soubry was a Remain supporter in the EU referendum, as was Cox.
