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.
The post read: "NO child deserves to be talked to in such a manner... He is a child, he is to be respected and he is off limits".
Chelsea Clinton, a former first child, joined the support for Barron.
SNL has made a number of shows in recent months, parodying President Trump.
Mr Trump, who took office on 20 January, has described the programme as "not funny" and with "terrible" cast.
A Saturday Night Live writer has been suspended after posting a tweet mocking US President Donald Trump's son Barron, US media reports say.