Both veteran players are attempting to end long victory droughts in the third of five tournaments that make up the PGA Tour's Fall Finish.
Mediate, who has not won since 2002, stayed hot with a 6-under 65 that gave him a 36-hole total of 129 at the CordeValle Golf Club. He made a hole-in-one Thursday while shooting an opening-round leading 64 and added an eagle at the par-4 fourth Friday.
Duval shot a 65 to jump into a tie for third place -- four shots behind Mediate.
The last of Duval's 13 tour victories came at the 2001 British Open, after which his career went into a decline. He was runner-up this year at the AT&T Pebble Beach Pro-Am, but that has been his only top-10 finish of the season.
Ryuji Imada was alone in second at 132 following a 67 and Dean Wilson shared third place with Duval. Rickie Fowler, who is looking for his first tour triumph, was part of a tie for fifth place at 134, along with Will MacKenzie, Henrik Stenson, Tim Herron and Bo Van Pelt.
Swim Week Miami Beach 2014 [PHOTOS]