"Our security team is investigating the veracity of these threats with relevant law enforcement officials," a Twitter spokesman said, without elaborating.
The news comes following reports that Twitter had suspended accounts linked to IS and related entities, including the Nigerian Islamic group Boko Haram.
A page on the online bulletin board Pastebin showed an image of Twitter founder Jack Dorsey in what appear to be the crosshairs of a rifle scope and a message in Arabic.
A translation of the message by SITE said that Dorsey and Twitter are now being targeted by Islamic militants for suspending certain accounts.
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"You began this failed war, and we have told you from the beginning that it is not your war! But you did not understand. You shut down our accounts and we quickly return, but when our lone lions shut down your breathing there will be no return!," the message said.
Like other social media operators, Twitter has struggled with allowing free speech while avoiding being used as a tool for violence.
But analysts note that terrorist organizations frequently use Twitter and other social media for recruiting, fundraising and communications.