Cold outreach with a connection in common:
Hi [name]
I noticed we're both connected with [name of connection]. How do you know [pronoun]? I also work in the [industry/niche] as a freelance writer!
OR
I noticed we're both connected with [name of connection]. Judging from your role at [company], we might be able to help each other out. Let me know if you'd like to connect sometime to talk shop!
Cold outreach without any connections in common:
Hi [name]
I came across your profile after seeing your post about [topic they posted about]. Interesting stuff...
what's your take on [related topic]?
OR
[compliment about blog post]
Note: When I'm working on content writing for clients, if I reference any blog posts, come across a helpful resource, or feature something that someone else created, I always go and connect with them using a message like this!
Someone who writes in your niche that would be a good connection but maybe not a client:
Hey [name]
I saw your recent post on [name of where you saw post]. The way you wrote about [subject] was [adjective]. I'd love to connect with you!
Note: This one is great for the same purpose as above.
To someone who recently secured a new role:
Hi [name]
I'm a freelance [niche] writer. I read about your new role (congratulations!) at [company] and I thought I'd get in touch to see if you'd like to connect here on LinkedIn.
Note: this can be used for cold outreach or warm outreach (like if you've already connected with someone else at that company)
To someone who recently viewed your profile:
Hi, [name]. I noticed that you took a look at my profile; would you like to connect?
Note: Every single time I send this message to someone who viewed my profile they end up asking to schedule a discovery call with me. It's worth the little effort and LinkedIn will let you know that someone viewed your profile!
To a manager of an org you want to work with:
Hi [name]
My name is [your name] and I am a freelance writer specializing in producing [niche] [service] for sites like yours. I've been enjoying the [service type] on [website name] and I was wondering if you'd like to connect.