Not the USA, but the CSA. The governor of Nuevo Leon and Coahuila, Santiago Vidaurri, asked the Confederacy to annex his northern Mexican state. He was incredibly powerful in northern Mexico, and likely could have also delivered Tamaulipas, Chihuahua and Durango as well. Northern Mexico was more connected economically to Texas than the rest of Mexico, and Vidaurri was wary of Benito Juarez attempting to reduce his near total control of the north.

Jefferson Davis turned down the offer as it would have cut off a potential route to export cotton (through Matamoros), leaving it open to union blockade. Also it would piss off the French, preventing a potential European recognition the Confederacy desperately wanted, as Napoleon III was openly preparing to intervene.

A good NYTimes article about the offer.

Are you looking for countries that could have been annexed, or countries interested in joining the United States willingly?

For the former, The Dominican Republic and the United States: From Imperialism to Transnationalism discusses how the United States considered annexing the Dominican Republic in the years after the U.S. Civil War but decided against it for fear of causing instability in the region.

Most countries in Latin America and the Caribbean have been considered for annexation at some point, through means as varied as William Walker's filibustering expeditions and the Spanish-American War.

For the latter, consider the short-lived Republic of Yucatan, which maintained a tenuous independence from Mexico in the first half of the 19th century. Someone better-informed than I can explain the history of that republic.

In 1847 and 1848, representatives from the Republic visited Washington, D.C. to request assistance in a nasty three-cornered civil war within the Republic. After their initial requests went unanswered, in desperation, they asked the United States to annex the Republic.

The U.S. House of Representatives approved a measure granting President James K. Polk permission to take control of the country, but the measure died in the Senate.

The debates regarding the issue can be found starting on Page 728 of the 30th Congress, 1st session of the Congressional Globe.

As /r/rumsodomy points out, there was some sentiment for formal secession in northern Mexico, which was never very fond of Mexico City. There were a number of filibusters and rebellions in the region before the U.S. Civil War: the most notable attempted to set up independent republics. The Republic of the Sierra Madre and the Republic of the Rio Grande both failed, but had they managed to gain traction it's possible they could have been annexed to the United States.

There was a very active annexation movement in Newfoundland in the late 40s, before the referendum that made it a Canadian province.

The U.S. government made quiet overtures to Denmark in 1946, offering to buy Greenland, unsuccessfully.

As someone mentioned in the thread, the Dominican Republic petitioned to join the USA, but the motion died in the Senate. By a single vote, if I remember correctly.

There was also brief discussion of Newfoundland joining the USA rather than Canada, but I don't think it was taken very seriously.

There were discussions with the Quebecois in 1775 to persuade them to join the American revolution, and they recruited soldiers who formed the 1st and 2nd Canadian regiments. They were one of the few portions of the army supported directly by congress. The rest were all supported by their particular states.

But strictly speaking it was not a country, per se.

Are you sure that you this blog referred to the USA and not the International Union of American Republics or other Pan-American organizations?