Both Barmby, 19, and Williams, 20, will stay with Pools for an initial month and will go straight into manager Colin Cooper's squad for Saturday's home league game against York, subject to Football League approval.
Barmby, who can play out wide or in a central role, is the son of former Liverpool and Tottenham midfielder Nick Barmby and has progressed through the Academy at Old Trafford into United's Under-21s. He is an England youth international.
Williams became Middlesbrough's youngest player in over 100 years when he replaced Jonathan Franks in a first-team match in December 2009 at the age of 16.
He made his full debut for Boro against Coventry in April 2010 and was named man of the match, but has struggled to force his way into Boro's first-team plans this season, being restricted to four appearances.