Norris' McLaren team-mate Daniel Ricciardo added he was "certainly considering" a social media blackout.
The Australian explained: "Obviously, if it helps, like all things, raise awareness and spread the message or put it out there, and hold someone with a bit more accountability, for sure, I think that’s important.
"When you grow up, if you don’t have anything nice to say, don’t say it at all, and sometimes it can be that simple. So, certainly under consideration and if we feel it’s going to help and have a positive effect then we’ll do so."
Williams driver and Grand Prix Drivers' Association director Russell has declared it his "duty" to take a stand against the abhorrent online abuse.
"I thought it was important to do so because there is far too much online abuse, hatred, negativity and racism that is just undeserved," he said.
"I feel like it is our duty to raise as much awareness as possible, not just within sport but within all walks of life. It doesn't matter how old or young you are or what background you are from, just to spread positivity and be nice to one and other.
"I feel strongly about this and I felt like it was my duty to be a part of this cause."
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