Chevron, the US oil firm fighting a Chinese rival for control of Unocal, has increased its offer and won the backing of the takeover target's board.
Chevron is looking to increase its presence in Asia
Chevron has now tabled a $63 (£36) per share bid in stocks and cash, up from an earlier bid of just under $60.
That values Unocal at $17.1bn, less than the $18.5bn all-cash bid put forward by Chinese oil company CNOOC.
Unocal's board has backed the Chevron bid after attempts by a Chinese firm to buy into the US proved controversial.
Chevron boss David O'Reilly said its new offer was "driven by competitive circumstances, but even at this higher price it remains a compelling transaction".
Any purchase by CNOOC would have to be examined by the President George W Bush's administration and there are concerns that the process could take months rather than weeks.
CNOOC has made repeated efforts to reassure politicians that its takeover would not jeopardise jobs or prove to be a security risk, and has given assurances that it would sell some assets and leave others in US hands.
"We think that our full-cash offer is still superior even after Chevron raised its bid," a CNOOC spokesman said.
China's fast-growing economy is increasing is oil imports rapidly.
And Unocal has a strong presence in developing countries, especially in Asia.
It operates in Thailand, Bangladesh, Indonesia, Myanmar, Azerbaijan, Congo and Brazil.
Unocal shareholders will get to vote on the bids at meeting on 10 August.