The threats follow a decision by the social networking site to block the accounts of jihadis.
"Our security team is investigating the veracity of these threats with relevant law enforcement officials," Efe news agency quoted the statement issued on Monday as saying.
On Sunday, people claiming links with the IS posted a threat against Twitter co-founder Jack Dorseyon on image-sharing site JustPaste, saying that his "virtual war" against the militants would trigger a real war against him.
The threat said in Arabic that "Twitter (is) a target for the caliphate".
"We told you from the beginning it's not your war, but you didn't get it and kept closing our accounts on Twitter. But we always come back," said supporters of the group that seeks to create an Islamic "caliphate" in large parts of Iraq and Syria.
The post went on to question how the Twitter founder would be able to protect the company's employees once they became targets of the militant group.
Twitter's user policy prohibits the posting of direct threats against other people.
Since last June, Twitter, together with other social networks, has been closing accounts of IS supporters.
Twitter was founded in 2006 and has more than 500 million users.