The bills, many of which will be backdated to the start of the year, could come in at up to €500 each.
Last night the company admitted the error and said that payment will be sought from customers in their next bill.
Speaking this morning Eircom's Director of Corporate Affairs Paul Bradley said: "It was very regrettable, it's embarrassing and we're very sorry that it's happened.
"We want to reassure people that this is not a question of over-charging. Money has not been taken out that should not have been taken out. Unfortunately we're in a position where we said that the bill was paid and actually, the money never came out of someone's account."
Eircom confirmed that customers will have to pay the full amount outstanding in one go.
She said: "We will not be crediting customers' accounts but we are always happy to work with customers who have any difficulties meeting their payments.
"This will be discussed with individual customers."
Mr Bradley told Newstalk's Pat Kenny that the average backdated payment will be about €100 for "the vast majority " of customers.
Eircom, which has two million customers, blamed a "systems error" for the failure to collect direct debit payments.
Affected customers include a mix of phone and internet users, mainly those with bundle packages, who make monthly direct debit payments.
Eircom told affected customers that it will reimburse a failed direct debit fee of €18.45 but refused to offer them a credit on their bills to compensate for the mistake.
The company blamed a new European payments system – the Single European Payments Area (SEPA).
In a statement it said: "We regret that, due to a system error related to the implementation of SEPA payment, up to 30,000 Eircom customers did not have some or all of their monthly direct debit payments taken from their bank accounts for phone, broadband and TV service since January."
Eircom said that customers had received bills that had incorrectly stated that the sums had been paid as usual from their accounts.
Once the error came to light, the company said it "immediately established a customer contact plan including phone calls and letters".
Saying that Eircom was still investigating how the problem occurred, the spokeswoman said: "We sincerely apologise for this error and can assure customers that this error has now been rectified."
The company has set up a special phone line for affected customers – 1800 303 432, which is open until 8pm, to deal with queries.