With world No 2 Rafael Nadal bringing down the curtain on the scheduled matches, taking on the United States' Donald Young, a few more chapters in the quest for this year's titles will be written by the end of the day.
Sixth seed Konta, 26, lost to 21-year-old Vekic, ranked 58, in the final of the Nottingham grasscourt tournament last month.
"It's another opportunity for me to try and find a way to come out on top this time," Konta told reporters about the match-up after her first-round win over Hsieh Su-Wei.
Twice champion Murray, who has been suffering with a sore hip, has already dispatched one player with an armoury of unconventional shots, Kazakh Alexander Bublik.
He now takes on 97th ranked Dustin Brown of Germany, whose trademark dreadlocks are often sent flying as he leaps for seemingly ungettable balls.
