Ladbroke's are Burnley's official United Kingdom betting partner, but took to Twitter to taunt their players.
'We have contacted Ladbrokes and are awaiting a full response'.
"It was only meant to be a bit of fun but failed completely in that regard, and we have apologised to Burnley FC and would like to sincerely apologise to any fans of the club that took offence".
Clarets manager Dyche claimed the abuse directed at the former Leeds duo was "strange" as he pointed out how "you make £20million for your club and you get slaughtered".
He said: "It's a odd business, football".
"Maybe I got the changes wrong, but I was expecting a lot more from my players".
It was doubly sweet for Leeds as Burnley fielded two players signed from Elland Road this year
Wood stepped up and slotted the first one for Burnley, but after each team had successfully netted three times, James Tarkowski's effort was saved, and Leeds - under the new ABBA penalties format - slotted two in succession to end up getting the final say. "But Woody has done fantastically, walking on to our pitch and scoring, and again in the shootout".
Chris Wood's late penalty levelled the scores in the third-round League Cup match.
The tie ended 2-2 after extra time.
"There's never a lack of will and demand from the group, my only gripe is the amount of opportunities we created", said Dyche.
The Poland worldwide was excellent in the middle of the park against the Premier League side as Leeds booked their place in the fourth round after a penalty shootout victory, and he may well now feature in many fans' lineup for the visit of the Tractor Boys at the weekend.
He also scored in the penalty shootout but Leeds, who are in a division lower, went through 5-3. I am very disappointed because this is a competition we wanted to do well in.