The pair will be available to the League One strugglers until the end of the season.
Martin Allen was desperate to bring in much-needed cover for his threadbare squad and will be pleased with his captures.
Berchiche is a Spanish youth international full-back, while Hutton is a Republic of Ireland U19 international winger.
The Robins have secured the services of the duo at no cost, with Spurs prepared to pay their wages during their time at Whaddon Road.
Both men will now be hoping to make their debuts against Walsall on Saturday.