The race to get a bitcoin exchange-traded fund in front of retail investors heated up Friday, when a firm run by technology entrepreneur Barry Silbert filed with the Securities and Exchange Commission to list its Bitcoin Investment Trust on the New York Stock Exchange.

The move by Grayscale Investments LLC a unit of Mr. Silbert’s Digital Currency Group Inc., could expand the audience for the fledgling virtual currency if the fund is approved by the SEC. It also could prove an early test for how an SEC run by a Donald Trump...