The 24-year-old agreed his £5.5million move to St James' Park last week, but a row over a £300,000 pay-off to which he was contractually entitled if he was sold without asking for a transfer has prevented the deal going through.
Barton's agent Willie McKay has referred the matter to the Professional Footballers' Association with City adamant that they will not pay because he was not forced out of the club.
Allardyce is being forced to play a waiting game, but has urged the Eastlands hierarchy to resolve the issue.
He told the Manchester Evening News: "They are out of order. This is a private matter, a confidential issue that should not have been made public. I'm flabbergasted and want the situation resolved.
"I am just trying as Newcastle manager to secure a player of this kind and quality and the deal has been held up.
"I cannot understand it because City didn't pay anything for this player as he came through the youth system and are selling him for over £5million.
"It's not really for me to comment. All I know is that this is a matter for Joey to sort out and I hope it is resolved as it has stopped him putting pen to paper."
Barton travelled to Tyneside last week to discuss his proposed move after both the Magpies and West Ham agreed a fee with City.
He was so impressed with what he heard and saw after meeting Allardyce, chairman Freddy Shepherd and prospective new owner Mike Ashley that he decided within hours to head for Newcastle without even talking to the Hammers.
However, although everything else is agreed, the outstanding payment, although unlikely to scupper the deal, has held things up.
Indeed, striker Mark Viduka, who met Newcastle officials in London a few days later, tied up his move within hours.
Barton was suspended by City at the end of last season following a training ground incident involving team-mate Ousmane Dabo which left his future in doubt.
However, Allardyce is confident he can harness the player to get the best out of him on the field.
He said: "I found Joey a very bright person who wants to develop his game.
"He is very positive, well-spoken and focused and wants to win silverware with Newcastle and develop his international career."