Alok Sharma, the conference chairman, urged the almost 200 national delegations present in Glasgow to accept a deal that seeks to balance the demands of climate-vulnerable nations, big industrial powers, and those whose consumption or exports of fossil fuels are vital to their economic development.
"Please don't ask yourself what more you can seek but ask instead what is enough," he told them, in the closing hours of a two-week conference that has already overrun by a day. "Is this package balanced? Does it provide enough for all of us?"
"Most importantly - please ask yourselves whether ultimately these texts deliver for all our people and our planet."
But before a plenary meeting could be convened to vote on the deal, delegates from India, China, the United States and the European Union met to discuss language on an agreed phase-out of coal, a member of the Indian delegation said.
The final agreement requires the unanimous consent of the countries present, ranging from coal- and gas-fuelled superpowers to oil producers and Pacific islands being swallowed by the rise in sea levels.
The meeting's overarching aim is to keep within reach the 2015 Paris Agreement's target to cap global warming at 1.5 degrees Celsius (2.7 Fahrenheit) above pre-industrial levels.
A draft deal circulated early on Saturday in effect acknowledged that existing commitments to cut emissions of planet-heating greenhouse gases are nowhere near enough, and asked nations to set tougher climate pledges next year, rather than every five years, as they are currently required to do.
In a public check-in round with key delegations, there was encouragement for Sharma when China, the world's biggest producer and consumer of coal, the dirtiest fossil fuel, said it had "no intention to open the text again".
The West African state of Guinea, which had pressed hard on behalf of the G77 group of developing countries for greater commitments from rich countries to compensate them for "loss and damage" from unpredictable climate disasters, also indicated that the group would accept what had been achieved.
However, India, whose energy needs are heavily dependent on its own cheap and plentiful coal, signalled unhappiness.
"I am afraid ... the consensus remained elusive," Environment and Climate Minister Bhupender Yadav told the forum, without spelling out whether or not India would block a vote on the package.
EU Climate Commissioner Frans Timmermans, speaking after Yadav, asked if the marathon conference was at risk of stumbling just before the finish line and urged fellow delegates:
"Don't kill this moment by asking for more texts, different texts, deleting this, deleting that."
Scientists say that to go beyond a rise of 1.5C would unleash extreme sea level rise and catastrophes including crippling droughts, monstrous storms and wildfires far worse than those the world is already suffering.
But national pledges made so far to cut greenhouse emissions - mostly carbon dioxide from burning coal, oil and gas - would only cap the average global temperature rise at 2.4 Celsius.
Saturday's draft, published by the United Nations, called for a phase-out of coal power as well as efforts to reduce the huge subsidies that governments around the world give to the oil, coal and gas that power factories and heat homes.
Previous UN climate conferences have all failed to single out fossil fuels for their harm to the climate.