Minister for Education Norma Foley has insisted that her officials are in "almost daily discussions with Bus Éireann" over school transport.
Ms Foley noted the "particular frustration" for children who had previously availed of concessionary tickets, but have been denied them this year.
She said that she is exploring whether there will be "scope" in Budget 2023 to help them.
Echoing remarks she made on RTÉ's Prime Time last night, Minister Foley said that she is in "very significant" discussions with Minister for Public Expenditure Michael McGrath "to secure the additional funding that would be required" in the Budget to help "that cohort".
"We'll have to see how that unfolds", she added.
Minister Foley was addressing the Joint Committee on Education, Further and Higher Education, Research, Innovation and Science.
Concessionary tickets are given to families who, while they do not qualify under the eligibility criteria, "are dependent" on the system, she said.
Some families have been refused a ticket this year, which is as a consequence of an "unprecedented increase" in demand from those who are eligible for the scheme, Minister Foley added.
The minister said that concessionary tickets "will be accommodated where there is capacity", and noted that there has already been an increase "of one third in concessionary tickets this year".
She also acknowledged "frustration over delays in issuing tickets" for children.
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Review due 'shortly'
The controversy "has dominated the headlines - especially in rural communities - over the last two months", Committee Chair Fine Gael TD Paul Kehoe said, opening the session.
Those people who are now being denied concessionary tickets are "the main reason" the minister is before the committee, Green Party Senator Pauline O'Reilly noted.
The eligibility threshold of "4.8km is simply too far for a secondary school for anyone", she added, insisting that it is not "viable" and must be reduced.
A "root an branch review" of the scheme is under way, and "every aspect is being looked at", Minister Foley said, adding that this involves "engagement with all of the stakeholders".
When pressed on when this review will conclude, she added, "I would be hopeful that it would complete shortly, by the end of this year".
However, the minister later added a qualification to hitting this target, stating that the department has "been impacted by some issues, most notably perhaps, Ukraine".
Senator O'Reilly said that her family was given a bus ticket, despite the fact that they no longer needed it - and did not even apply for it.
Another senator, Fianna Fáil's Fiona O'Loughlin, said that she has heard of other families being given tickets which they did not apply for.
While Sinn Féin's Rose Conway-Walsh said that the whole debacle "could have been avoided with proper budgeting and proper planning".
She accused the minister of "deflecting responsibility" while children, who are "collateral damage", are "left at the side of the road".
'Absolute evidence'
Paul Kehoe said "there's a huge amount of anecdotal evidence", on top of which he has "absolute evidence", of abuse of the system.
Four pupils he knows have "absolutely no intention" of using their bus tickets, he said.
"That's happening right across the country", Deputy Keogh claimed.
"I think it was a huge mistake that there wasn't some form of nominal fee", he added.
In an acrimonious exchange, Minister Foley said Deputy Keogh should provide his information to the authorities. It is not his job "to police", he retorted.
Scheme expanded
More than 124,000 students are now using the scheme which is managed by Bus Éireann on behalf of the Department of Education.
An expansion of the scheme in July resulted in a jump of more than a fifth using the scheme on this time last year, where the figure was closer to 100,000.
Across the whole of the 2021/22 school year, more than 121,400 children availed of school transport, including more than 15,500 with special educational needs, the minister said.
The cost in 2021 was €289m.
There are "just over 600 pupils" for whom Bus Éireann is still trying to secure "buses or bus drivers", the minister said, but insisted that those pupils will get the transport they are entitled to.
Sinn Féin TD Donnchadh Ó Laoghaire said that the department "should have seen this coming", and "prepared to find the additional capacity over the summer months".
"You build your life around the school run in the morning", he said, and those people who had access to school transport, and are now being denied it, "feel particularly let down".
Deputy Ó Laoghaire said that urgent action was needed.
Independent TD Carol Nolan said that the Government's decision to waive fees for school transport for this year is welcome, but suggested that it should have come in tandem with substantially enhanced capacity.
