Moors travel north of the border to face Lowland Football League outfit Kelty Hearts on Saturday, while Bryan Hughes’ men host Ayr United of the Scottish Championship that same afternoon.
Both sides will be looking to end a difficult run of form in recent League matches, with both only being able to record draws in midweek.
The two highest finishing Clubs remaining in the Vanarama National League 2018/19 season were invited into the Competition by the SPFL at the Third Round stage, just four victories away from the Final.
Invited Clubs from NIFL Premiership, Welsh Premier League, and League of Ireland Premier Division were also selected to compete.
At the time of the draw, National League director Sid Blain said: "It's a great honour for The National League to be involved in the Scottish Challenge Cup once again.
"Last season, both Sutton United and Boreham Wood performed very well, and didn't concede a goal in any of their matches.
"Solihull Moors and Wrexham will be strong candidates going into the competition this campaign," he added.
Goal updates will be available on The National League’s social media channels on Saturday afternoon.