The SPL rivals have been paired together in the draw for the quarter-finals, which will take place across the first weekend in March.
Elsewhere Celtic must overcome their League Cup semi-final conquerors St Mirren on away territory for a last-four spot.
And the semi-finals will definitely include a team from the First Division after the two remaining non-SPL sides - Hamilton and Falkirk - were drawn to meet at New Douglas Park.
Kilmarnock will host Hibernian in the last of the quarter-finals.
William Hill Scottish Cup quarter-final draw:
Hamilton v Falkirk
Dundee v Dundee United
Kilmarnock v Hibernian
St Mirren v Celtic