By Arathy Somasekhar

HOUSTON (Reuters) -Oil prices rose about $2 on Monday, bolstered by supply fears, a dip in the U.S. dollar and early strength in equity markets, but prices seesawed as some worried fuel demand could weaken if the Federal Reserve raises U.S. interest rates too aggressively.

Brent crude futures for September settled up $1.95, or 1.9%, at $105.15 a barrel, while U.S. West Texas Intermediate (WTI) crude futures rose $2, or 2.1%, to settle at $96.70 a barrel.

"A slightly weaker U.S. dollar and improving equity markets are supporting oil," UBS oil analyst Giovanni Staunovo said. After early strength, U.S. stocks moved lower in afternoon trading, with investors cautious about the Fed meeting this week and earnings from several growth companies.

Oil futures have been volatile in recent weeks, pressured by worries that rising interest rates could slow economic activity and fuel demand but supported by tight supply, especially since Russia's invasion of Ukraine and Western sanctions on Moscow.

"The U.S. and European economies are slowing and with the Federal Reserve set to raise interest rates again this week, traders remain very cautious," said Dennis Kissler, senior vice president of trading at BOK Financial.

Fed officials have indicated the U.S. central bank would likely raise rates by 75 basis points at its July 26-27 meeting.

China, the world's second-biggest economy, narrowly missed a contraction in the second quarter, growing just 0.4% year-on-year.

But a steep front-month premium over the second month continues to signal near-term supply tightness. The spread settled at $4.82/bbl on Friday, an all-time high when excluding expiry-related spikes in the two previous months.

Libya's National Oil Corporation (NOC) said it aimed to bring back production to 1.2 million barrels per day (bpd) in two weeks, from around 860,000 bpd.

But analysts expect Libya's output to remain volatile as tensions remained high after clashes between rival political factions over the weekend.

Prices drew support from "expectations that Russian oil supply will edge lower in the months ahead as widely-expected plans for a price cap on Russian oil may have the opposite effect on oil prices than hoped for," said Warren Patterson, head of commodities strategy at ING.

The European Union said last week it would allow Russian state-owned companies to ship oil to third countries under an adjustment of sanctions agreed by member states last week aimed at limiting the risks to global energy security.