They quote a person familiar with SNL's plans as saying Katie Rich was suspended indefinitely. The programme's network, NBC, is yet to comment.
On Friday, Ms Rich posted a now-deleted tweet, saying "Barron will be this country's first home-school shooter".
An avalanche of criticism on social media followed.
Ms Rich later deleted her tweet, suspending her Twitter account.
However, on Monday she was back online to issue an apology.
One Facebook post in reaction to her previous tweet has been shared nearly three million times.