Plaid Cymru has confirmed for the first time that talks are underway with Labour over the Welsh government’s budget.

The party says it has made a “public offer” to do a deal, on the basis more cash is found for councils and the health service.

Party leader Rhun ap Iorwerth said the Welsh government’s existing plans, announced in October, would “hit our public services hard” .

The Welsh government said it was “committed to working with all parties in the Senedd to pass the budget”.

Plaid’s leader accused the First Minister, Eluned Morgan, of not knowing what was going on with the budget talks as the two clashed in the Senedd on Tuesday.

At the last budget Labour ministers were able to rely on the single vote of Jane Dodds, the Welsh Liberal Democrat leader, to get their spending plans through a Senedd vote.

But Plaid’s victory in the Caerphilly by-election has meant that Labour needs two extra votes to get next January’s budget through the Senedd.

To find a deal, ministers have already spoken to the Welsh Conservatives, Dodds and independent MS Russell George.

Although Plaid has struck budget agreements with Labour in the past, it joined the Conservatives in voting against the last set of spending plans.

Meanwhile the Conservatives say they have further meetings planned between their Senedd leader, Darren Millar, and Eluned Morgan.

Last October draft budget published by Finance Secretary, Mark Drakeford, planned to increase each Welsh government department’s funds rise by around 2%.

But the figure is expected to change, because the budget left hundreds of millions unallocated and available for discussions with opposition parties on how they might want to spend it.

There were warnings that if the budget did not change, NHS organisations and councils may need to cut services and jobs, when costs are rising at different rates in different services.

Drakeford has said he is open to providing extra money to both services through the budget talks.

A Plaid source confirmed that the party’s finance spokesperson, Heledd Fychan, and Drakeford had met.

They said the figure of extra cash to work with was not precisely known – it was reported to be £230m for day to day services in early November.

Eluned Morgan came under fire from Rhun ap Iorwerth in the Senedd on Tuesday, claiming that thousands of council jobs were on the brink.

“A grim picture emerges of public services teetering on the edge,” he said, saying that the spending increase for the NHS amounts to 0.3% in real terms.

Morgan told the Senedd on Tuesday that “our door is open” to Plaid.

“We have invited you to talk to us about the budget… you could help to fix that.”

In response, Ap Iorwerth told the first minister to “keep her fingers on the button and keep up with what’s going on”.

“Plaid Cymru is, first minister, ready to get Wales out of the hole made by Labour,” he said.

A later statement from him accused Morgan of being “seemingly unaware that initial discussions had already taken place to understand the budgetary situation”.

Ap Iorwerth said Morgan had “completely ignored the public offer by Plaid Cymru to allow a budget settlement to save the NHS and councils from the harm of Labour’s original budget tabled”.

The Welsh Conservatives have called for scrapping stamp duty on all main homes in their talks with the Welsh government.

Welsh Conservative leader in the Senedd Darren Millar said: “Another Labour and Plaid Cymru-backed budget deal will mean yet more taxpayers’ money squandered on vanity projects, rather than the priorities of the people of Wales.

He added: “We continue to engage, in good faith, in the budget negotiation process with further meetings scheduled.”

By Gareth Lewis, BBC Wales political editor

No one seems in any doubt that a deal will be done, because of the threat of huge cuts to public services if it is not.

But it’s not done until it’s done.

The Welsh government will view today as Plaid publicly accepting that there cannot be a “no deal scenario”.

Plaid will try to portray this as a serious attempt to mitigate the effects on public services of a budget they fundamentally disagree with.

Rhun ap Iorwerth’s party might well be running those public services from next May, depending on the result of the Senedd election.