President Obama’s push to expand trade with a cluster of mid-sized Asian nations is couched in terms of increasing U.S. jobs and exports, but there’s a larger aim: forcing China to speed the opening of its economy and curbing some of the country’s economic clout.

Approaching his second term as China inaugurates its own new leadership team, Obama is pushing deeper into a “pivot” toward Asia that aims to strengthen security ties with U.S. allies in the region and reshape the American presence across the continent.

The complex U.S.-China relationship — partly cooperative, intensely competitive and of global importance — has made trade a central part of that strategy as the administration tries to build an economic bloc that would virtually encircle China with U.S. economic allies.

The effort received a boost during Obama’s current Asia trip when Thailand announced it hopes to join the Trans-Pacific Partnership (TPP), an under-construction free-trade agreement that includes several nations on China’s southern frontier, as well as others elsewhere in the region and in South America.

Officials from Vietnam, Malaysia, Singapore and other TPP nations are scheduled to meet with Obama in Cambodia on Tuesday to discuss the wide-ranging trade agreement. The session occurs just before Obama holds his last face-to-face talks with outgoing Chinese leader Wen Jiabao.

The administration is trying to complete the TPP talks in the coming year. If successful, it will present China’s new leaders with a potentially large, U.S.-influenced trading bloc emerging on their home turf.

“This isn’t about marginalizing China, but about defining the agenda” for economic policy in the region, said Mireya Solis, a senior fellow at the Brookings Institution and associate professor at American University.

While U.S. officials tout the potential export benefits of the TPP, National Security Council spokesman Ben Rhodes said ahead of Obama’s trip that it is also about “advancing our values as well as our interests . . . . We see extraordinary potential in deepening the economic ties within the Asia Pacific.”

Individually, the eight nations involved with the United States in the TPP are of modest economic interest. The United States already has trade agreements with several of them, including as Singapore and Australia. The others are mid-sized nations whose potential to buy U.S. exports, absorb American investment or provide low-priced goods for the American market is dwarfed by the roughly half-trillion dollars in goods and services that moves between the United States and China each year.

Trade between the United States and Thailand, for example, amounted to only $34 billion in 2011 — with Thai shrimp, seafood and locally assembled electronics moving in one direction, and U.S. wheat, semiconductors and industrial chemicals and equipment moving in the other.

But the list of TPP countries is growing. Mexico and Canada are negotiating to join, making preferential access to the giant North American market a prize for any other country that joins. Japan, the world’s third-largest economy, behind China, has also expressed interest. The United States already completed a free-trade agreement last year with South Korea, an important American political and economic ally to China’s north. The addition of Thailand would extend the TPP’s potential sweep along China’s southern border.

Successful conclusion of the TPP is not guaranteed. Some civil society groups oppose the pact, arguing that it may weaken drug access in less developed members of the TPP, or lead to more offshoring of U.S. manufacturing jobs.

The agreement itself, however, is being negotiated squarely with China in mind — either as a potential member, or as a nation that may have to respond to the rules and policies set by the TPP nations.

The talks, approaching their 15th round next month in New Zealand, address many old-line trade concerns, as countries such as Australia push to get more of their locally grown sugar into the United States, and Vietnam lobbies to further open the U.S. market to its textile and footwear products.

But the talks are far broader than many of the bilateral agreements the United States has struck in recent years. They are delving into issues such as patent protections for pharmaceutical companies, regulation of the Internet, the opening of financial and other service sectors for U.S. companies and how to set clear rules for the management of state-owned enterprises.

Many of those cutting edge questions involve issues in which the United States has clashed with China and pushed for reform. U.S. officials hope that establishing clear standards on those issues within the TPP might provide incentive for China to change — if companies, for example, begin favoring Vietnam or Malaysia for new investments because of the access it gives them to the United States and other TPP nations.

“They want to make this the free-trade area of the Asia-Pacific,” said Scott Miller, a senior adviser at the Center for Strategic and International Studies. “You have some meaningful economies in it now that are willing to accept the obligations . . . . If China wants to follow the standards there is no reason they could not join.”