Hojgaard needs to finish in a two-way tie for 29th or better in the tournament being held at The Belfry to earn enough points to overtake Shane Lowry, who is clinging onto sixth place in the list of automatic qualifiers.
Hojgaard is currently in a tie with five other players — including his twin brother, Nicolai — for 12th place, three shots off the lead shared by Marcel Siem, Li Haotong, Matthias Schwab and Thomas Aiken.
The top six get into the European team automatically. Rory McIlroy, Justin Rose, Tommy Fleetwood, Robert MacIntyre and Tyrrell Hatton are already guaranteed to be at next month's matches at Bethpage Black in New York, but Lowry isn't competing at The Belfry.
Luke Donald will announce his six captain's picks to complete the 12-man team on September 1.
One of those is expected to be Matt Fitzpatrick, who sat one shot off the lead on Thursday after seven birdies and two bogeys in an opening 67.
Among the Australian contingent at The Belfry, David Micheluzzi and Elvis Smylie led the way after one-under 71s, putting them in joint-22nd position. Jason Scrivener and Danny List were a shot further back on level par.
