Who sponsored an attempt to settle Virginia with English colonists in 1587?

The most likely answer to your question is Sir Walter Raleigh.  Raleigh was the man who recruited and sent a group of colonists to Virginia in 1587.  However, depending on what you mean by “sponsored,” the answer could be Queen Elizabeth I.
Queen Elizabeth was the ultimate sponsor of the colonization of Virginia.  In 1584, she gave Raleigh a charter giving him the right to colonize North America.  She stipulated that he had to establish a colony or lose the right to do so.  Raleigh sent some colonists out in 1585 to create such a colony.
By 1587, however, the colony was in bad shape.  Most of the colonists had returned to England.  At that point, Raleigh sent out another group of colonists.  These settled in an area that was then called Virginia, but which is now in North Carolina.  Raleigh was not able to send any further ships to contact the colony until 1590.  When the ship arrived, it found no one left in the former colony.  This colony has come to be known as the Lost Colony.
So, we can say that Sir Walter Raleigh sponsored this expedition as he was the man directly in charge of making it happen.  However, we could also say that Queen Elizabeth sponsored it as she gave Raleigh the right to send the expedition to Virginia.
https://www.history.com/news/archaeologists-find-new-clues-to-lost-colony-mystery