The short answer is there is no historical evidence for any sort of substantial or prolonged contact between Rome and the Americas; that is to say, there is no surviving account of an expedition to there and back again, nor any significant, confirmed material evidence of trade or exchange between the two regions prior to the Norse colonization at Vinland in the 10th century. There are some discussions on Pre-Columbian Contact in the FAQ, but nothing specifically discussing Romans.

The slightly longer answer is that there has been substantial interest in possible pre-Columbian transatlantic contact between Europe and the Americas for decades, and there are some isolated artifact discoveries which are ambiguous but might suggested one-time or occasional contact. A good example of this kind of thing is the "Roman figurine" discovered at Calixtlahuaca - warning, the page will make your eyes bleed, but the significant point put forward by Dr. Michael E. Smith there is:

The “Roman figurine” supposedly excavated at Calixtlahuaca was not documented using standard archaeological procedures. Excavator José García Payón did not publish professionally adequate descriptions of any of his excavations at the site. After his death, two posthumous reports were issued (García Payón 1979; 1981), but these contain very little specific information on the excavations or individual contexts. The “Roman figurine” cannot be considered well documented according to the normal standards of archaeological practice. [...] These problems of data reporting affect more than just the “Roman figurine” from Calixtlahuaca. The lack of documentation applies to nearly all of the finds from García Payón’s fieldwork. While these problems do not invalidate the “Roman figurine” as a potentially valid Precolumbian find, their implication is that it is impossible today to reconstruct the archaeological context of the find. It certainly cannot be claimed that this find is “well documented” or that it comes from “a good archaeological context.” The excavation of the “Roman figurine” fails to meet even the minimum standards of archaeological reporting.

This is really more of an archaeological point - identifying artifacts and their proper context and documenting the process of discovery can be difficult and frustrating - but it bears repeating here because its exactly the situation we have to deal with when looking at the claims here. Historians have to follow a similar process when they evaluate evidence, trying to trace the provenance of document evidence and evaluating its reliability.

For the "amphorae" discovered in the Bay of Guanabara in Brazil in particular, the best version of the popular account is Treasure Lost at Sea: Diving to the World's Great Shipwrecks by Robert and Jennifer Marx. Robert F. Marx is well-known as a diver - and also a treasure-hunter who has made several claims about pre-Columbian contact with the Americas, and author or co-author of a number of non-fiction works on the subject. The details of the jars claim, however, are not verifiable - there's been no formal, documented excavation or survey of the underwater site; no published results of independent examination of the artifacts; nor to the best of my knowledge has Marx presented his finds to academic publication - although he did publish "Romans in Rio?" (PDF) in Oceans in 1984. The Fringe of American Archaeology: Transoceanic and Transcontinental Contacts in Prehistoric America (PDF) claims at least one account that an attempt by an expert in Roman antiquities to view the jars was denied. Even assuming Marx was operating in good faith - and his moderate claim of a single Roman ship-wreck rather than any sort of ongoing exchange stands out in his father - the lack of documentation of the find, and inability for independent analysis and confirmation of the recovered jars, makes them unreliable as a point of evidence.

Did the Romans have ships capable of crossing the Atlantic?