Prices have been driven up by production curbs in OPEC nations and Russia, as well by strong demand thanks to healthy economic growth, Reuters said.
Read alsoExpert says Ukraine's gas reserves enough for 22 years of extractionBrent crude futures LCOc1 dipped 26 cents, or 0.4 percent, to $70 per barrel by 0624 GMT on Tuesday from the previous day's close. But traders said Brent was well supported overall around this level.
Brent hit $70.37 a barrel on Monday, a December 2014 high, when markets were at the beginning of a three year-long slump.
U.S. West Texas Intermediate (WTI) crude futures CLc1 were at $64.48 a barrel, up 18 cents, or 0.3 percent from their last settlement. WTI hit a December-2014 peak of $64.89 a barrel in early trading.
Oil has been pushed higher by an effort led by the Organization of the Petroleum Exporting Countries (OPEC) and Russia to withhold production since January last year. The cuts are set to last through 2018.
The restraint has coincided with healthy oil demand, pushing up crude by almost 15 percent since early December.
"This rally has been driven first by robust fundamentals, with strong demand growth and high OPEC compliance accelerating," U.S. bank Goldman Sachs said in a note on Tuesday.
"We see increasing upside risks to our $62 per barrel Brent and $57.5 per barrel WTI forecast for the coming months," it added.
Other U.S. banks, including Bank of America Merrill Lynch and Morgan Stanley, have already upped their price forecasts.
"We have updated our supply/demand balances to reflect a faster-than-expected tightening in the global oil market due to improving cyclical conditions, cold winter weather, and higher than expected OPEC compliance," Bank of America Merrill Lynch said.
"We now see a deficit of 430,000 barrels per day (bpd) in 2018 compared to 100,000 bpd prior, and thus see Brent crude oil prices averaging $64 per barrel in 2018 compared to $56 prior. Our WTI projection also moves up from $52 per barrel to $60 per barrel for the same reasons," the bank said.
Morgan Stanley said "oil markets were 0.5 million bpd undersupplied in 2017," adding that 2018 would still see a 200,000 bpd deficit.
It expected Brent to rise to around $75 per barrel by the third quarter of this year.
A major factor holding back crude prices in 2017, the surge in U.S. production, has stalled at least temporarily as icy winter weather in North America has shut down some facilities.
Instead of hitting 10 million bpd this month, as widely expected, U.S. production C-OUT-T-EIA fell from 9.8 million bpd in December to 9.5 million bpd currently.
However, most analysts still expect U.S. production to break through 10 million bpd soon.