The alleged threat came in a social media debate around a billboard in Banbridge.
Mr O’Dowd said he has reported the matter to police.
Earlier his Upper Bann running mate Liam Mackle complained that his party’s election posters have been stripped from Banbridge, and stolen in Portadown and Lurgan.
He said he is seeking a meeting with police about the thefts.
“Banbridge town has been practically stripped of Sinn Fein posters, in Portadown masked Loyalists ripped posters from poles and Lurgan has also seen significant losses,” he said.
“The sheer scale of theft and destruction of Sinn Fein posters in Upper Bann means the police must respond to protect democracy.”
Responding to the complaint about stolen posters, Chief Inspector Lynne Corbett said police investigate all incidents reported to them.
“Police in Armagh, Banbridge and Craigavon are committed to doing everything we can to ensure all candidates can participate fully in the forthcoming Northern Ireland Assembly election free from harassment, intimidation and other forms of criminality,” she said.
“We are investigating thoroughly all incidents which have been reported to us, including the removal and destruction of election posters, and any identified suspects will be liable to prosecution.
“I would be happy to engage with any person or party with concerns on this matter and would appeal to anyone who witnesses the theft or damage of election posters to contact police immediately.”
It is the latest in a number of complaints of election posters being removed ahead of the Assembly poll on May 5.
Police previously revealed they had received dozens of reports about election posters being damaged, removed or destroyed across Northern Ireland during this election campaign.
Earlier this month SDLP South Belfast candidate Elsie Trainor said she was attacked by two youths as she confronted them over the removal of some of her posters in the Ormeau Park area.