Morley and Outwood MP Andrea Jenkyns posted the "appalling" email she received on Sunday night on Twitter, insisting: "We will not let these people beat us down. Going to keep fighting for Brexit."
The post was seen by the Metropolitan Police's contact centre, which said officers in the relevant department would be making contact with Ms Jenkyns to arrange for a report to be filed.
The Conservative MP has been one of the most vocal advocates for the Brexit demands of the European Research Group (ERG) led by arch-Eurosceptic Jacob Rees-Mogg since quitting her role as a Government aide in May to speak out on the issue.
In recent days she has submitted a letter of no confidence in Prime Minister Theresa May.
If 48 Tory MPs put in letters a leadership challenge is triggered.