No.7 seed Djere beat Spaniard Roberto Carballes Baena 7-5 7-6 (7-1) on the hard surface, while No.8 seed Kwon comfortably saw off Japan's Yoshihito Nishioka 6-1 6-2.
Elsewhere, Thiago Monteiro earned an entertaining 6-2 3-6 7-6 (7-2) win over Daniel Altmaier, and the Brazilian will now face No.3 seed Marin Cilic in the Round of 16 after the 2014 US Open champion was given a first round bye.
The other encounter on Monday saw Steve Johnson come through in three sets against wildcard Aleksandar Vukic, 6-4 2-6 6-4. The American will go up against No.2 seed Karen Kachanov in the last 16.
Two-time grand slam semi-finalist Gael Monfils is the No.1 seed for the ATP 250 tournament, with Cilic, Khachanov and Frances Tiafoe among the other big names who will enter the competition in the next round.