Some elements within the US did in fact argue for the annexation of Mexico towards the end of the Mexican-American war. It's commonly referred to as the All-Mexico movement and it was an important component of Washington politics during the ratification of the Treaty of Treaty of Guadalupe Hidalgo. The diary of James Polk, sitting president of the time gives us some idea of the influence of this idea in the political climate of the war's end:^1

I cannot help laboring under the conviction that the true reason of Mr. Buchanan's present course is that he is now a candidate for the Presidency, and he does not wish to incur the displeasure of those who are in favour of the conquest of all Mexico.

However, those who advocated for complete Mexican annexation like Buchanan and Edward Hannegan faced a huge amount of opposition.^2

Most notable among the opposition was John C. Calhoun, who opposed the annexation on racial grounds. He delineated the conquest of Mexico as fundamentally different from the conquest of Indian territory on the basis that Indians had been killed or driven out by white settlers. With the burgeoning population of Mexico, this strategy would not be remotely practicable.

I know further, sir, that we have never dreamt of incorporating into our Union any but the Caucasian race—the free white race. To incorporate Mexico, would be the very first instance of the kind of incorporating an Indian race; for more than half of the Mexicans are Indians, and the other is composed chiefly of mixed tribes. I protest against such a union as that!

...

Are we to associate with ourselves as equals, companions, and fellow-citizens, the Indians and mixed race of Mexico? Sir, I should consider such a thing as fatal to our institutions.

Calhoun was not alone in these sentiments. Many Americans had varying reasons to oppose the complete annexation of Mexico. The task of integrating the primarily Catholic, hispanic country promised to mongrelize the US. Indeed, Manifest Destiny itself was conflicted on the issue of annexation, given its roots in the idea of white American superiority.

Overall, the All-Mexico movement never gained enough influence to stop the Treaty of Guadalupe Hidalgo, the passage of which quickly killed the idea. Its adherents found significant resistance to a supposed mongrelization on racial, social, and religious grounds. Not only that, but some political leaders at the time (Thomas Benton being the primary example) opposed the entire war itself, arguing the annexation did not serve the interests of the country.

^1 James K. Polk, The Diary of a President 1845-1849, Allan Nevins, Ed. (New York: Longmans, Green, and Co., 1952)

^2 "Religion without Restriction": Anti-Catholicism, All Mexico, and the Treaty of Guadalupe Hidalgo John C. Pinheiro Journal of the Early Republic, Vol. 23, No. 1 (Spring, 2003), pp. 69-96

In 1807 Aaron Burr, the former vice president under Thomas Jefferson, was put on trial for treason by Jefferson. He was accused of planning on "separating the western states from us, of adding Mexico to them, and of placing himself at their head." Burr had been in the new western territories, gathering allies and leading a small group of armed men down the Mississippi River. Burr's exact scheme will never be known, but Jefferson thought it involved capturing New Orleans and invading Mexico from there.

Burr was acquitted for lack of evidence.