The "Equality Calling" campaign began delivering messages to federal politicians on Monday.

Organised by Australian Marriage Equality (AME), it allows supporters of same-sex marriage to record a voice message on a 1300 number that is then delivered straight to their locals MPs and senators.

But on Monday, after receiving the first batch of recordings, one of Mr Andrews' electorate office staff members contacted the group to ask that the messages stop.

Fairfax Media understands the staff member reasoned that the Defence Minister  - who is known for his conservative views - believed marriage should be between a man and woman - and would not change his position.

He also said the calls were clogging up the phone lines. According to AME, only two messages have come in for Mr Andrews so far. Together they last for about five minutes.

One is from a woman, who says she is approaching 30, and tearfully explains how she wants to be able to marry her female partner, before thanking Mr Andrews for listening.

The other, which the office has not listened to, is from a primary school teacher who wants to marry his partner of 29 years.

"We just want the right to be able to publicly celebrate our relationship in front of our family and friends," he said.

"Quite frankly, we're getting a bit tired of going to all these other weddings and watching everybody else celebrate their love for each other.

"And we're not allowed to do that ... We're not getting any younger and we want to do this before it is too late."

According to a spokeswoman for the Defence Minister, his electorate office received a "number of automated calls".

"His office contacted the organisation concerned and expressed that further contact be made in writing to best facilitate a personal response."

AME national director Rodney Croome said it was "very disappointing" that Mr Andrews was refusing to listen to his constituents.

"Politicians should welcome engagement with their voters, not actively deny those voters a voice.

"It would be entirely inappropriate to hang up on a constituent calling in real time. And it's just as inappropriate to hang up on a constituents's heart-felt recorded message."

Mr Croome said that at the end of the voice messages, it was also possible for MPs to opt out from receiving further messages, but said they would still be posted on social media "for all the world to see what they refuse to acknowledge".

National Party senator Matt Canavan's office has also opted out of the calls.

A spokesman for Senator Canavan said that it was "well versed with the issue".

He explained that the initial calls were helpful, but the small office found the ongoing calls "distracting" and time consuming.

The spokesman said that people were still able to ring the office directly on a range of issues, including same-sex marriage.

AME said that Labor's defence spokesman Stephen Conroy - who did not vote for same-sex marriage in September 2012 - had also opted out of the messages, however a spokesman for Senator Conroy said this had been an inadvertent mistake.

"Senator Conroy is happy to receive all communications from the public on any issue," he said.

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