Glasgow Tory MSP Annie Wells said she has contacted Police Scotland over the social media posts.
She told the Scottish Mail on Sunday she has been advised to increase her security by varying her way to work.
Ms Wells, who has previously spoken of coming to terms with coming out as gay as a young teenager, said the majority of the abuse on Twitter was aimed at her simply for a being a woman.
She told the paper that in the three years since she was elected as a list MSP for the Glasgow region, she has been forced to mute more than 1,100 abusive Twitter accounts due to the violent and foul-mouthed nature of the posts.
She said: “There was one tweet in particular which threatened violence. He was tweeting that he knew exactly where I was. I told Police Scotland and they put a mark on my house and surgery.
“I also told security at Holyrood and they told me to use a different route to work and not to go out by myself.
“After I received that particular tweet, I was due to hold a surgery in a room in Govan Library. I used to have a room in the back, but now I have it in the public part of the library.
“When we were in the library, I feared we could be violently attacked. That is the impact these tweets have had.”
A spokesman for the Scottish Parliament told the paper: “Regrettably, elected representatives face increasing threats and abuse, both online and in person.
“Over the past two years, the parliament’s Corporate Body has provided numerous security enhancements to local offices based on Police Scotland advice and it has made lone worker devices available to MSPs and their staff.
“We continue to work closely with Police Scotland in offering MSPs personal safety advice and support.”
Police Scotland said it was committed to “ensuring the safety and security of all of our communities, including MPs, MSPs and elected officials”.
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