Owners of Mila, a Bedlington Terrier, found her cowering in a bush in their Halesowen back garden on Thursday evening.
Burglars beat the dog - leaving her with nerve damage - after breaking into the house, West Midlands Police said.
Homeowner Julie Green said she was "heartbroken" to find Mila wounded by burglars, who did not steal anything.
"Whoever is responsible is heartless," said Ms Green.
"She is the friendliest dog and would greet anyone. Even though she is poorly she has been trying to roll over and wag her tail."
Vets believe Mila has neurological nerve damage, preventing her from placing her front feet on the ground.
Ch Insp Phil Dolby said police are following several lines of inquiry.
He added: "This did not happen in the dead of night - there will be people around the Albrighton Road area who could have seen the offender or offenders responsible for the break-in."
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Burton, 62, worked briefly as Albion coach under Roy Hodgson before returning to the club in May 2014, to work alongside the club's sporting and technical director Richard Garlick.
Day, 60, recruited in July 2014, was with Albion less than a year.
"It was felt changes were required," said Albion chairman Jeremy Peace.
Since appointing Garlick in January 2013 as 'sporting and technical director' to replace the FA-bound Dan Ashworth, Albion have had four head coaches - Steve Clarke, Pepe Mel, Alan Irvine and Tony Pulis.
"We are looking ahead positively to a new beginning under Tony Pulis after two gruelling seasons," Peace told the Albion club website. "All our focus now is on preparing the squad for a stronger performance.
"I am on record as saying we would be engaged in an on-going review at all levels. In Tony, we have a very experienced and hands-on operator."
Former West Ham keeper Day was head of scouting and recruitment at Championship side Brighton prior to join Albion, while Burton left his post as Arsenal's head of youth coaching to return to the Midlands.

The 31-year-old failed a drugs test following their 36-8 defeat by Wakefield Trinity on 14 July.
The Super League club said they had received notification from UK Anti-Doping and the Rugby Football League.
A Widnes statement said: "The club fully respects this suspension and are currently reviewing this matter."
Chase won the Man of Steel award in 2011 - a year in which he would later go on to represent England and earn eight caps.
He joined Widnes from the Tigers in May, initially on loan, before the deal was made permanent in July.
BBC Sport's rugby league correspondent Dave Woods
Rangi Chase has been the ultimate enigma since he arrived at Castleford eight years ago. Unbelievably skilful, he has produced some of Super League's most breathtaking moments.
But equally breathtaking has been his often off-field stupidity. His has been a career full of official warnings, last chances and shown doors by a collection of clubs that have lost patience with his extra-curricular activities.
He's come and gone at Cas, Salford, Leigh, Cas again and now, presumably, Widnes. If this latest drug test failure is upheld, there will finally be no way back in rugby league for his mercurial, magical but misguided talents.
Precedent has been set. Gareth Hock, then at Wigan, was banned for two years when he was found guilty of cocaine misuse ahead of a game.  If the same holds for Chase, that would take him to 33. Age alone would probably be against him. But reputation will hold even greater sway.
A premature end to his career, if that's what this is, will surprise no-one but it will sadden many that such skills, such sublime ability that he possesses couldn't have been harnessed for the even greater pleasure of all those who have marvelled at the very best of Rangi down the years.

6 October 2015 Last updated at 08:13 BST
The nine-day event will host more than 500 balloons, representing 20 different countries.
Thousands of visitors are expected to enjoy the spectacular sights.
An 18-month-old family dog is unable to walk after burglars repeatedly kicked it in the head, police said.
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West Bromwich Albion technical director Terry Burton and head of recruitment Mervyn Day have left The Hawthorns as part of an expected summer reshuffle.
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Widnes Vikings half-back Rangi Chase has been suspended from all training and playing activity after testing positive for cocaine.
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The 44th annual Albuquerque International Balloon Fiesta is under way in the US.