The United Arab Emirates' energy minister said that Opec members were ready to reduce output, the Wall Street Journal reported.
Venezuela's oil minister said oil-producing nations were on a "very good path" to clinch a deal.
However, traders said sharp falls on Thursday may have triggered some bargain-hunting.
Eulogio Del Pino, the Venezuelan minister, who recently visited Russia and Saudi Arabia as part of a global tour to drum up support among both Opec and non-Opec producers, said "we're on a very, very, very good path" to reducing production.
Brent crude closed up $3.30 at $33.36 a barrel in New York after falling below $30 on Thursday.
After sinking to a 12-year low of $26.05 on Thursday, US crude settled up 12%, or $3.23, to $29.44 a barrel - its biggest one-day rise since 2009.
Many traders were sceptical about the Journal's report, pointing out that Venezuela and Russia had tried in vain earlier this week to stir Saudi Arabia and other major producers into agreeing to output cuts.
However, some believe that prices would rebound sooner or later if production tightened or demand rose.
Commerzbank analysts said: "We expect declining US oil production, in particular, to drive the oil price back up to $50 per barrel by the end of the year."
Some traders still expected wilder price swings in the coming weeks. "It's not a one-way price movement anymore" in oil, said ABN Amro's senior energy economist Hans van Cleef. "We will see a period of high volatility".
Friday's price rises were also aided by figures from oil services company Baker Hughes, which said that US energy firms cut the number of oil rigs for the eighth consecutive week to the lowest levels since January 2010.
Drillers removed 28 oil rigs, bringing the total rig count down to 439, Baker Hughes said.
The jump in oil prices helped to boost sentiment on stock markets, with the FTSE 100 in London closing 3.1% higher at 5,707 points.
Wall Street was also trading higher on Friday, with the S&P 500 rising 1.8% and the Dow Jones Industrial Average up close to 2% in late trading.
Oil prices surged as much as 12% on Friday after new suggestions that Opec nations were set to cut oil production.