The Immigration Department is struggling to cope with a massive influx of inquiries about the government's changes to the citizenship test, with callers turned away or placed in lengthy queues.

Audio recorded on Monday and obtained by Fairfax Media shows clients are being disconnected and told to "call back later", or otherwise placed in queues of close to 300 people.

It comes after the Turnbull government announced a major overhaul of the citizenship test on Thursday, including the addition of a "values" test, tougher English language requirements and a four-year wait as a permanent resident.

"Due to very high demand we are unable to take your call at this time," worried callers are being told by a machine. "The department apologises for any inconvenience and we thank you for your patience. Please call back later."