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."

The governing body took the decision to switch the club's membership to OK Bulls Ltd after Khan's group bought the club from the administrators in August.
Bradford's membership was suspended in June when they entered administration.
"The board's decision closes the door on much of the uncertainty that has surrounded the Bulls in recent months," RFL director Blake Solly said.
"The club's new owners have co-operated fully with the RFL throughout this necessarily extensive process and we look forward to working with Mr Khan and Mr Sutcliffe in the months and years ahead.
"The decision to grant membership is an important step that will help the board to determine which competition Bradford Bulls will compete in from 2013."
No decision has yet been reached on what division the Bulls will compete in next season but an announcement is expected by the end of the week on how that decision will be reached.
Bradford finished ninth in Super League this season, three points outside the play-offs, having been deducted six points for entering administration.

The case revolved around a horn "hit", which was allegedly lifted from the Salsoul Orchestra track Love Break.
But the US 9th Circuit Court of Appeals said the sample lasted less than a second, and would not have been recognisable to the general public
"Without careful attention, the horn hits are easy to miss," it said.
Producer Shep Pettibone worked on both Vogue and Love Break. The one-note horn sequence in contention lasted just 0.23 seconds, the court heard.
You can hear the original at 4'40" in this clip, and at 1'02" in Madonna's track.
"After listening to the recordings," wrote judge Susan P Graber, "we conclude that a reasonable jury could not conclude that an average audience would recognise the appropriation of the composition".
However, the court's decision was not unanimous, with Judge Barry G Silverman arguing that the uncredited sample, if proven, would amount to theft.
"It is no defence to theft that the thief made off with only a 'de minimis' part of the victim's property," Silverman wrote.
He said a copyright of a recording amounted to a "valuable property right, the stock-in-trade of artists who make their living recording music and selling records".
The ruling could lead to short samples becoming more commonplace.
However, as The Hollywood Reporter points out, the California court's decision directly contradicts a 2006 case presented in the 6th Circuit (Tennessee).
That revolved around an NWA song that sampled a riff from George Clinton's band Funkadelic. At the time, a 6th Circuit judge wrote: "Get a license or do not sample. We do not see this as stifling creativity in any significant way."
Police have released recordings of a suspected burglar phoning for a taxi from the house he had just broken into.
[NEXT_CONCEPT]
The takeover of Bradford Bulls by Omar Khan's consortium has been ratified by the Rugby Football League.
[NEXT_CONCEPT]
Madonna's Vogue did not break copyright law, even though it contained a snippet of another artist's song, a US court has ruled.