Described in writings of the era as a Venetian jeweler living in Agra at the time, the idea of a European designer first seems to have been proposed shortly after the British occupation of Delhi and Agra in 1803.
One article notes:
The first person to suggest this theory of an Italian Architect of the Taj Mahal was a Spanish monk by the name of Father Sebastian Manrique, an Augustinian Friar whose purpose in India was to secure the release of Father Antony, who was being held as a prisoner by the Mughals in Lahore.
There were a few small elements — jewel inlays within the mausoleum and minor floral reliefs in the marble — that could have possibly been influenced by Italian design, although in whole, the entirety of the Taj Mahal overwhelmingly bears more resemblance to Islamic aesthetics and Indo-Persian design.
Author Jack S. Dixon in “The Veroneo Controversy” makes the point that “apart from technical and historical arguments, one need only look at the great Islamic buildings in India to reject any thought that Europeans had any hand in their design, construction or embellishment.”
So, Veroneo is just one more of the many myths surrounding the Taj Mahal.