The man dialled two taxi firms from the landline in the house at Wodecroft Road, Luton, but hung up after he was asked for his phone number.
Bedfordshire Police asked anyone who recognised the man's voice to contact them.
A flat-screen television and other electrical items were stolen from the house.
The calls were made in the early hours of Saturday 25 October using a cordless telephone, which was then left on a table in the garden, police said.
The man called two Luton taxi firms which were both dealt with by the same control office.
In the first call, the suspect was asked to leave his landline number, but said he did not know it.
He was then asked to leave his mobile number and he started to, but then stopped, claiming he did not know it.
Police said he then made a second call and got through to the same controller, who explained the office handled calls for more than one taxi firm.
The force hopes the suspect, who almost revealed his mobile phone number, can be identified from the recordings, which were made automatically.
Det Con Scott Hannam said: "We are keen to identify the man as we believe he may have important information which could assist our investigation.
"I would urge anyone who recognises his voice, or for the man himself, to contact police.
"I am also keen to hear from anyone who has any information or saw a large television being carried around this area on the night in question."
Police have released recordings of a suspected burglar phoning for a taxi from the house he had just broken into.