I'm a strong believer in early and continuous (interrupted) contact between the 'new' and 'old' worlds (tens of thousands of years). This question has baffled me for a long time. Now, recently I've read an article that pushes the date of the arrival of TB in the Americas back by 1,000 years.
1. John Day's letter
'It is considered certain that the cape of the said land was found and discovered in the past by the men from Bristol who found Brasil, as your Lordship well knows. It was called the Island of Brasil and it is assumed and believed to be the mainland that the men from Bristol found.' (From the letter written by the English spy John Day between mid-December 1497 and mid-March 1498 to "The most magnificent and most worthy lord, the Lord Grand Admiral" in Spain (likely Christopher Columbus). The letter was discovered in the Spanish archives in 1955 by the American researcher Dr. Louis Andre Vigneras. This English translation published in Canadian Historical Review, volume XXXVIII (1957), pages 219-228).
2. Sir Francis Bacon:
'..it is likely that the discovery first began where the lands did nearest meet. And there had been before that time a discovery of some lands, which they took to be islands, and were indeed the continent of America, towards the north-west. And it may be, that some relation of this nature coming afterwards to the knowledge of Columbus, and by him suppressed (desirous rather to make his enterprise the child of his science and fortune than the follower of a former discovery), did give him better assurance that all was not sea from the west of Europe and Africke.'
(Fragment from Sir Francis Bacon's Historie of the Reigne of King Henry the Seventh [written in 1621, published in 1638]).