Armagh must make the trip to Westmeath while Cavan are at home to Tipperary.
Monaghan were beaten by Down in the Ulster semi-finals on Saturday while Wexford saw off Limerick in the first round of qualifiers.
McEnaney guided Monaghan to the 2007 and 2010 Ulster finals, losing to Tyrone on both occasions.
Monaghan will be favourites to get past Wexford despite playing away, while Armagh should fancy their chances of seeing off a Westmeath team heavily beaten by Dublin at Croke Park on Sunday.
Wexford and Westmeath were promoted from Division Four of the Football League earlier this year.
Wexford manager Seamus McEnaney will face his old county Monaghan in the second round of the All-Ireland Championship qualifiers.