United's 2-2 draw with Well last Sunday added to their fixture congestion but kept alive their hopes of a successful defence of the Scottish Cup.
The replay has been scheduled for March 29, meaning Scotland assistant boss Houston will return to prepare his United side for the match immediately after the clash with Brazil at the Emirates Stadium.
Speaking ahead of Wednesday`s Clydesdale Bank Premier League clash with Hearts, Houston said: "We're planning for the 29th because there's nothing else to go for.
"We haven't got dates for it, so unless Motherwell look to change it... but I don't know when we'll be able to fit it in."
United are in the midst of a run of five games in 12 days, all at home, with Hearts the opposition on Wednesday and Inverness travelling to Tannadice on Saturday.
Source: PA
Source: PA