WASHINGTON—The Ecuadorean government declined late Monday to say whether it had cut off internet access to WikiLeaks founder Julian Assange, but said it would continue providing him asylum, as they have in their London embassy since 2012.

Early Monday, WikiLeaks announced on Twitter that Mr. Assange’s internet access had been “severed by a state party.” Several hours later, it alleged that Ecuador was behind the action, prompting a response from Quito.

The...