A representative for Specific Media could not be reached for comment. Golden Gate Capital declined to comment.
The deal is likely to be a mix of cash and stock for less than $100 million.
While Specific Media and Golden Gate Capital are close contenders, other bidders are involved in the process including Myspace co-founder Chris DeWolfe in a partnership with Austin Ventures; Criterion Capital Partners, which bought social media site Bebo from AOL; and Activision Blizzard Chief Executive Officer Bobby Kotick, according to the source.
Additionally, more than 50 percent of Myspace's 500-strong workforce is expected to be laid off because of the sale.
The deal is expected to be sealed on the heels of the Sun Valley conference, the annual gathering at an Idaho resort of media and technology moguls, as well as a host of other U.S. and international luminaries.
News Corp paid $580 million in 2005 for Myspace -- once a pioneer of the social networking space -- but soon lost ground to Facebook, now the world's No. 1 social networking site.
Last year, News Corp relaunched Myspace as a social entertainment site with a focus on music, movies and celebrities.