The most significant price action this week is likely to take place after the release of the American Petroleum Institute's weekly inventories report on Tuesday and Wednesday's U.S. Energy Information Administration report.
Despite the drop in the rig count, which is often used an as early indicator of future production, activity remains much higher than a year ago when just 617 rigs were active.
US West Texas Intermediate (WTI) crude futures were at US$61.91 a barrel at 0755 GMT, down 13 cents, or 0.2 per cent.
Brent crude futures were at $65.11 per barrel, down 38 cents from their previous close.
Iran wants OPEC to work to keep oil prices around $60 a barrel to contain USA shale producers, Oil Minister Bijan Zanganeh told The Wall Street Journal in a rare interview, while Saudi Arabia has played down shale's ability to upset the market and has indicated $70 for oil is acceptable.
2 dead after helicopter crashes in NYC's East River
He made a mayday call to LaGuardia Airport saying that the aircraft experienced engine failure. "East River - engine failure". The New York City Fire Department tweeted that its members were with police at the location near 91st Street in Manhattan.
Oil prices fell on Monday as investors grappled with ongoing concerns over rising U.S. output and tight Opec supply, while last week's data showing speculators cut bets on oil suggested more selling could be seen. That split is driven by differing views over whether $70 a barrel sends US shale companies into a production frenzy that could cause prices to crash.
The rising USA output comes largely on the back of onshore shale oil production.
"Oil prices moved lower. after (the) Energy Information Administration published a report that crude production from seven major US shale plays is expected to see a climb", said Stephen Innes, head of trading for Asia/Pacific at futures brokerage OANDA in Singapore. At stake is OPEC's production limits, which are among factors helping the oil market's monthslong recovery. It was the first such decline in seven weeks.
In other trading, April gasoline fell 0.5% to $1.894 a gallon, while April heating oil fell 1.2% to $1.865 a gallon.
Although crude's value has risen over 20% in the last six months, oil futures have been buffeted by a short-term bearish prognosis as USA output has revved up and crude stocks have swelled to record highs.