It follows the first formal talks since Carwyn Jones and Leanne Wood were tied 29-29 in a Senedd vote on Wednesday.
The aim was to form a minority Labour administration rather than a coalition, a joint statement on Friday said.
"We are confident that we can find a way forward that will result in the successful nomination of a First Minister next week," it said.
"We will continue to work over the weekend, and look to resume formal talks on Monday."
The Conservatives and UKIP, had joined Plaid Cymru in backing Ms Wood in the vote for first minister, with Labour and the sole Liberal Democrat, Kirsty Williams, backing Mr Jones.
Jane Hutt represented Labour and Simon Thomas Plaid Cymru in Friday's talks, which took place at Welsh Government offices in Cardiff Bay, with neither party leader attending.
Labour fell short of a majority at the 5 May election and needs opposition support to govern.
Plaid Cymru AMs insist they did not do a deal with other parties before the tied vote on Wednesday.
Welsh Conservative leader Andrew RT Davies told BBC Radio Wales on Friday there had been three sets of "substantial" talks involving two AMs from his party and two AMs from Plaid.
But he later released a statement suggesting the Tories would drop their backing for Ms Wood if a deal with Plaid was not forthcoming.
"We have an historic opportunity to break the mould in Wales... but this will only be possible with clear enthusiasm from others," he said.
"Should such ambition not be shown... Welsh Conservatives will not be minded to repeat our action from Wednesday's vote."
UKIP assembly group leader Neil Hamilton said the average Plaid voter did not back the party to see it "effectively" go into coalition with Labour.
"We are all in favour of talks, but minority parties should stick together against Labour arrogance," he said.
On Thursday, some UKIP AMs suggested they could switch their votes if parts of their manifesto were made law, although Mr Gething said for Labour: "That is not something we would consider."
Meanwhile, Labour and the Lib Dems confirmed talks were continuing between their parties.
BBC Wales understands Mr Jones has discussed appointing Ms Williams as a cabinet minister, but neither party would confirm any details of the talks.
Even with Ms Williams on board, Labour would still need an arrangement with another party to secure Mr Jones's re-election as first minister.
At the assembly election Labour won 29 seats, Plaid Cymru 12, the Conservatives 11, UKIP seven and the Liberal Democrats one.
AMs have until 1 June to nominate a first minister or face another election.
Former first minister Rhodri Morgan has suggested Welsh Labour could learn lessons from the SNP in the way the two parties reacted to last week's elections.
Nicola Sturgeon's SNP was two seats short of a majority in the Scottish Parliament, with Welsh Labour winning 29 of the 60 seats in the Senedd.
Speaking on BBC Radio Cymru's O'r Bae programme Mr Morgan said: "Nicola Sturgeon was saying this is the third victory in a row that we've had in Scotland.
"So you create a psychology, so that everyone expects that you will master, you will govern, and that nobody can ask a question about what the way forward is.
"Now Labour was saying, no, don't rejoice, don't claim that we've won, because we haven't.
"Technically Nicola Sturgeon hadn't either, but she had created the psychology of expectation."
Labour and Plaid Cymru say they are "confident" of breaking the deadlock over the election of a first minister.